fulton-harris-representation-theory (1991)

William Fulton Joe Harris Representation Theory AFirst Course Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest William Fulton Joe Harris Department ofMathematics Department ofMathematics University ofChicago Harvard University Chicago,IL60637 Cambridge,MA02138 USA USA Editorial Board JH.Ewing FW. Gehring PR. Halmos Department of Department of Department ofMathematics ‘Mathematics ‘MathematicsIndianaUniversity University ofMichigan UniversityofSantaClara Bloomington,‘Ann Arbor, Santa Clara,IN47405USA ML48109USA CA95053USA ‘Mathematies Subject Casifcation: 20605, 7BY0, 17820, 22E46 LibearyofCongressCataloging-in- PublicationDataFulton, Wiliam, 1939-Representation theory:estcourseWiliamFultonandJoeHarts.'p._cm.— (Graduate texts inmathematics) Inelades bibtiographiealrelerences andindex |.Representationsofgroups. 2.Representationsofalgebras 3,Liegroups, 4.Liealgebras, 1,Hares Joe. TL.Till UL Seren. QAITLESS 191S12.2-de09 90-24926 Printed onacid-ee paper (©1991 Springer-Verlag New York Ine. {Allrighis reserved. This work may notbetranslated orcopied inwhole orjnpat without thewritten permission ofthepublisher Springer-Verlag New York, Inc, 175Fith Avenve, "New York,NY10010,USA),exceptfrbritexcerplsinconnectionwithreviewsorscholarly analy. Use inconection with aay form ofinformation storage and retrieval, electronic‘daplation,computersofware,orbysimilarordissimilarmethodologynowknownorhereafter Seveloped isforbidden i ‘Theuseofgeneral descriptive names, rade name, trademarks, et,inthis pobifeafion, even iftheformerarenolespecialyieniidottobetakenaasignthaluchnames,arunderstood bytheTrade Marks and Merchandige Marks Act, may accordingly beused rely byanyone. “Typeset byAsco Trade Typesetting, Lid,Hong Kong Printed andbound byR.R. Donneliey &Sons Co, Harrisonburg, VA. Printed inthe United States ofAmerica. par6saszt ISBN0-387.97495-4 Springer-Verlag NewYorkBerlinHeidelberg(softcover)SBN3-540-97495-4 Springer.VerlegBerlinHeidelbergNewYork "SBN 0-387-97527-6 Springer-Verlag New Vork Berlin Heidelberg (hardcover)SBN3-540:97527-6 Springer.VerlagBerlinHeidelbergNewYork Preface ‘The primary goal ofthese lectures istointroduce abeginner tothefinite- dimensional representations ofLiegroups andLiealgebras. Since thisgoal is sharedbyquiteafewotherbooks,weshouldexplain inthisPrefacehowour approach differs, although thepotential reader canprobably scethisbetter byaquick browse through thebook. Representation theory issimple todefine: itisthestudy oftheways in which agiven group may actonvector spaces. Itisalmost certainly unique, however, amongsuchclearlydelineated subjects, inthebreadth ofitsinteresttomathematicians. Thisisnotsurprising: groupactionsareubiquitousin20th century mathematics, and where theobject onwhich agroup acts isnota ‘vector space, wehave learned toreplace itbyonethatis(e.g., acohomology. Broup, tangent space, etc). Asaconsequence, many mathematicians other than specialists inthefield (oreven those who think they might want tobe) come incontact with thesubject invarious ways. Itisforsuch people that this text isdesigned. Toputitanother way, weintend this asabook for beginners tolearn from andnotasareference. “This idea essentially determines thechoice ofmaterial covered here. As ‘simple asisthedefinition ofrepresentation theory given above, itfragments considerably when wetrytogetmore specific. Forastart, what kind ofgroup Garewedealing with—a finite group likethesymmetric group &,orthegenerallineargroupoverafinitefieldGL,(F,),aninfinitediscretegrouplikeSL,(Z), aLiegroup likeSLC, orpossibly aLiegroup over @localficld?Needlesstosay,eachofthesesettingsrequiresasubstantially differentapproach toitsrepresentation theory. Likewise, what sortofvector space isGactingon:isitoverC,R,Q,orpossibly@fieldoffinitecharacteristic? Isitfinite dimensional orinfinite dimensional, and ifthelatter, what additional structure stich asnorm,orinnerproduct)doesitcarry?Variouscombinations. vi Pretace ofanswers tothese questions lead toareas ofintense research activity inrepresentation theory,anditisnaturalforatextintendedtopreparestudents foracareer inthesubject tolead uptooneormore ofthese areas. AS@corollary,suchabooktendsogetthroughtheelementary materialasquicklyaspossible: ifonehasasemester togetuptoandthrough Harish~Chandra modules, there islitle time todawdle over therepresentations of©,and SLC. Bycontrast, thepresent book focuses exactly onthesimplest cases: repre~ sentations offinite groups and Liegroups onfinite-dimensional real and complex vector spaces. This isinsome sense thecominon ground ofthe subject, thearea that istheobject ofmost oftheinterest inrepresentation theory coming from outside. ‘The intent ofthis book toserve nonspecialists likewise dictates tosome degree ourapproach tothematerial wedocover. Probably themain feature‘ofourpresentation ithatweconcentrateonexamples,developingthegeneral ‘theory sparingly, andthen mainlyasausefulandunifyinglanguagetodescribe phenomena already encountered inconcrete cases. Bythesame token, wefor themost part introduce theoretical notions when andwhere they areuseful foranalyzing concrete situations, postponingaslongaspossiblethosenotions that areused mainly forproving general theorems. Finally, ourgoal ofmaking thebook accessible (ooutsiders accounts in part forthestyle ofthewriting. These lectures have grown from courses of thesecond author in1984 and 1987, and wehave attempted tokeep theinformalstyleoftheselectures.Thusthereisalmostnoattemptatefficiency:‘where itseems tomake sense fromadidacticpointofview,weWorkoutmany special cases ofanidea byhand before proving thegeneral case; and we cheerfully give several proofs ofonefactifwethink they areilluminating. ‘Similarly, whileitiscommontodevelopthewholesemisimplestoryfromone pointofview,saythatofcompactgroups,orLiealgebras,oralgebraicgroups, wehave avoided this,aseficientasitmaybe. Itisofcoursenotastrikinglyoriginalnotionthatbeginnerscanbestlearn about asubject byworking through examples, with general machinery onlyintroduced slowlyandastheneedarises,butitseemsparticularly appropriatehere. Inmost subjects such anapproach means onehasafewoutofan unknown infinity ofexamples which areuseful toilluminate {hggeneralsituation. Whenthesubjectistherepresentation theoryofcomplexsemisimple Liegroups andalgebras, however, something special happens: once onehas ‘worked through alltheexainples readilyathand—the“classical”casesofthe special linear, orthogonal, and symplectic groups—one has notjust afew useful examples, onehasallbutfive“exceptional” cases This isessentially what wedohere. Westart with aquick tour throughrepresentation theoryoffinitegroups,withemphasisdetermined bywhatisusefulforLiegroups.Inthisregard,weincludemoreonthesymmetric groupsthan isusual. Then weturn toLie groups and Liealgebras. After some preliminaries and alook atlow-dimensional examples, and one lecture with Pretace vii some general notions about semisimplicity, wegettotheheart ofthecourse: ‘working outthefinite-dimensional representations oftheclassical groups.ForeachseriesofclassicalLiealgebrasweprovethefundamental existence theoremforrepresentations ofgivenhighestweightbyexplicitconstruction. Our object, however, isnotjust existence, buttoseetherepresentations inaction,toseegeometricimplications ofdecompositions ofnaturallyoccurring representations, andtoseetherelations among them caused bycoincidences between theLiealgebras. ‘The goat ofthelatsixlectures istomake abridge between theexample- oriented approach oftheearlier parts andthegeneral theory. Here wemake anattempt tointerpret what hasgone before inabstract terms, trying tomake ‘connectionswithmodernterminology. Wedevelopthegeneraltheoryenough toseethat wehave studied allthesimple complex Liealgebras with fiveexceptions. Sinoetheseareencountered lesfrequentlythantheclassicalseries,itisprobably notreasonable inafirstcoursetoWorkouttheirrepresentations asexplicitly,althoughwedocarrythisoutforoneofthem.Wealsoprovethegeneral Weyl character formula, which canbeused toverify andextend manyoftheresultsweworkedoutbyhandearlierinthebook.Ofcourse, thepoint wereach hardly touches thecurrent state ofaffairs in Lietheory, butwehope itisenough tokeep thereader's eyes from glazing ‘over when confronted with alecture that begins: “Let Gbeasemisimple Liegroup, Paparabolic subgroup, ...”Wemight also hope that working, throughthisbookwouldpreparesomereaderstoappreciate theelegance(and cliciency)oftheabstractapproach. Inspirit thisbook isprobably closer toWeyt's classic [Wet] than tootherswrittentoday.Indeed,asecondary goalofour book istopresent manyofthe results ofWeyl and hispredecessors inaform more accessible tomodern readers. Inparticular, weinclude Wey!’s constructions oftherepresentationsofthegeneralandspeciallineargroupsbyusingYoung'ssyminetrizers; andweinvoke @litle invariant theory todothecorresponding result forthe orthogonal andsymplectic groups. Wealso include Weyl's formulas forthe characters ofthese representations interms oftheelementary characters of symmetric powers ofthestandard representations. (Interestingly, Weyl onlygavethecorresponding formulasintermsoftheexteriorpowersorthegenerallinear group. Thecorresponding formulas fortheorthogonal andsymplectic groups were only given recently byKoike andTerada. Weinclude asimple new proof ofthese determinantal formulas) More about individual sections can befound inthe introductions toother parts ofthebook. Needless tosay, aprice ispaid fortheinefficiency and restricted focus of these notes. The most obvious isalotofomitted material: forexample, we include little onthebasic topological, differentiable, oranalytic properties of Liegroups, asthisplaysasmallroleinourstoryandiswellcoveredindozens ofothersources,including manygraduatetextsonmanifolds. Moreover,therearenoinfinite-dimensional representations, noHarish-Chandra orVerma wilt Preface modules, noSteifel diagrams, noLiealgebra cohomology, noanalysis on symmetric spacesorgroups,noarithmeticgroupsorautomorphic forms,and. nothing about representations incharacteristic p>0.There isnoconsistent attempt toindicate which ofourresults onLiegroups apply more generally toalgebraic groups over fields other than Ror€(eg, local fields). And there isonly passing mentionofotherstandardtopics,suchasuniversalenveloping, algebras orBruhat decompositions, which have become standard tools ofrepresentation theory.(Expertswhosawdraftsofthisbookagreedthatsometopic weomitted must notbeleftout ofamodernbookonrepresentation theory—butnotwoexpertssuggestedthesametopic.) Wehave nottried (0trace thehistory ofthesubjects treated, orassign credit, ortoattribute ideas tooriginal sources—thisisfarbeyondourknow!- edge. When wegive references, wehave simply tried tosend thereader to sources thatareasreadable aspossible foroneknowing what iswritten here. ‘Agood systematic reference forthefinite-group material, including proofs of theresults weleave out, isSerre [Se2]. For Liegroups and Liealgebras, Serre [Se3], Adams [Ad], Humphreys [Hu!}, and Bourbaki [Bour} are recommended references, asaretheclassics Weyl [Wel] and Littlewood Tui).‘Wewouldlike(othankthemanypeoplewhohavecontributed ideasandsuggestions forthismanuscript, amongthemJ-F.Burnol,R.Bryant,J.Carrell, B.Conrad, P.Diaconis, D.Eisenbud, D.Goldstein, M.Green, P.Griffiths, B.Gross, M. Hildebrand, R.Howe, H.Kraft, A.Landman, B.Mazur, N.Chriss, D.Petersen, G.Schwartz, J.Towber, andL.Tu.Inparticular, we would like tothank David Mumford, from whom welearned much ofwhat ‘wekntow about thesubject, andwhose ideas arevery much inevidence inthis, book. Had thisbook been written 10years ago, wewould atthispoint thank the people who typed i.That being nolonger applicable, perhaps weshould thank instead theNational Science Foundation, theUniversity ofChicago,andHarvardUniversity forgenerously providingthevariousMacintoshes onwhich this manuscript was produced. Finally, wethank Chan Fulton for making thedrawings. BillFulton anilJoeHarris Using This Book ‘fewwordsareinorderaboutthepracticaluseofthisbook.Tobeginwith,prerequisites areminimal: weassume only abasic knowledge ofstandard first-year graduate material inalgebra and topology, including basic notions about manifolds. Agood undergraduate background should bemore than enough formost ofthetext; some examples andexercises, andsome ofthe discussion inPart IVmay refer tomore advanced topics, butthese canreadily beskipped. Probably themain practical requirement isagood working knowledge ofmultilinear algebra, including tensor, exterior, andsymmetric products offinite dimensional vector spaces, forwhich Appendix Bmay help. ‘Wehave indicated, inintroductory remarks (oeach lecture, when anyback- ground beyond thisisassumed and how essential itis.Foracourse,thisbookcouldbeusedintwoways.First,thereareanumber oftopicsthatarenotlogicallyessentialtotherestofthebookandthatcan beskimmed orskipped entirely. Forexample, inaminimal reading onecouldskip$$4,5,6,11.3, 134,153-155,173,19.5,20,221,223, 233-234,25., and 26.2;thismightbesuitableforabasicone-semester course.Ontheotherhand, inayear-long course itshould bepossible towork through asmuch ofthe material asbackgroundandjorinterestsuggested.Mostofthematerialinthe Appendices isrelevant only tosuch atong course. Again, wehave tried {oindicat, intheintroductory remarks ineach lecture, which topics are inessential and may beomitted. ‘Another aspect ofthe book that readers may want toapproach indifferent‘waysistheprofusion ofexamples. Theseareputinlargelyfordidacticreasons:‘wefeelthatthisisthesortofmaterial thatcanbestbeunderstood bygainingsomedirecthands-onexperiencewiththeobjectsinvolved.Forthemostpart,however, they donotactually develop newideas; thereader whose tastes run‘moretotheabstractandgeneralthantheconcreteandspecialmayskipmany x Using This Book ofthemwithoutlogicalconsequence. (Ofcourse,suchareaderwillprobablywind upburning thisbook anyway.)Weincludehundredsofexercises,ofwildlydifferentpurposesanddifficulties.Some aretheusual sorts ofvariations oftheexamples inthetext orarestraightforward verifications offactsneeded;«studentwillprobablywanttoattemptmostofthese.Sometimes anexerciseisinsertedwhosesolutionisaspecialcaseofsomething wedointhetextlater,ifwethinkworkingonitwillbbeuseful motivation (again, there isnoattempt at“eliciency,” andreadersareencouraged togobacktooldexercisesfromtime(otime).Manyexercises,areincluded that indicate some further directions ornewtopics (orstandard topics wehave omitted); abeginner may best beadvised (0skim these for ‘general information, pethaps working outa fewsimple cases. Inexercises, we {ried toinclude topics that may behard fornonexperts toextract from the literature, especially theolder literature. Ingeneral, much ofthetheory isintheexercises—and mostoftheexamplesinthetext.Wehave resisted theidea ofgrading theexercises by(expected) difficulty, although a“problem” isprobably harder than an“exercise.” Many exercisesarestarred:the+isnotanindication ofdifficulty,butmeansthatthereadercan find some information about itinthe section “Hints, Answers, and References” attheback ofthebook. This may beahint, astatement ofthe answer, acomplete solution, areference (owhere more can befound, or @combination ofanyofthese. Wehope these miscellaneous remarks, as, haphazard and uneven asthey are,will beofsome use Contents Preface ¥ Using This Book ix Part I:Finite Groups 1 1.Representations ofFinite Groups 3 §L.L: Definitions 381.2:CompeteReduiiliy:Schor's Lemma 5$1.2:Examples:AbelianGroups 8 2.Characters 12 §2.1: Characters 2 §2.2: The First Projection Formula and ftsConsequences 1s $23: Examples: and My i §2.4: More Projection Formulas; More Consequences: a 3.Examples; Induced Representations; Group Algebras; RealRepresentations 26 $3.1: Examples: S,and 9%, 6 $32: Exterior Powers oftheStandard Representation ofS, 4 §3.3: Induced Representations 2 §3.4:TheGroupAlgebra % §3.5: Real Representations and Representations over Subfields ofC0 sii Contents 4,RepresentationsofG,:YoungDiagramsandFrobenius’s Character Formula “4 $4.1: Statements ofthe Results “4§42:Irreducible Representations ofS, 32643: ProofofFrobenivs's Formula s 5,Representations ofM,andGL3(F,) 6$5.1:Representations of e$52:Representations ofGL(,)andSLa(F,) o 6.Weyl's Construction 75 $64: Schur FunctorsandTheirCharacters 15 $62: The Proofs cy PartHzLieGroupsandLieAlgebras 89 7.LieGroups 93 1.4: LieGroups: Definitions 3 {972 ExamplesofLieGroups 95 $7. Two Constructions 01 8.LieAlgebrasandLieGroups 10448.1:LieAlgebras:Motivation andDefinition 104$82. ExamplesofLieAlgebras, mt §8.: The Exponential Map 4 9.Initial Classification ofLieAlgebras 120 $9.1:RoughClassification ofLieAlgebras 121 $9.2 Enge's Theorem andLie's Theorem 125 {@. Semisimple LieAlgebras 128 $94: Simple LieAlgebras Bt 10.LieAlgebras inDimensions One, Two, andThree 133 $1041: DimensionsOneandTwo 133 §102: DimensionThree,Rank1 136 $102: Dimension The, Rank 2 139 $104: Dimension Three, Rank 3 1a 11.Representations ofsl,€ 146§11.1:TheIreducible Representations 146$11.2:ALittlePlethysin 151$113: ALittle Geometric Plethysm 153 Contents xii 12,Representations ofst,€,PartI 161 13.Representations ofs{,€,PartIl:MainlyLotsofExamples 175 $13.4: Examples 175, §132: Description ofthe Irreducible Representations 182 §133: ALittle More Plethysm 185 §134: ALittle More Geometric Plethysm 189 Part I1l:‘TheClassical LieAlgebras andTheir Representations 195 14,TheGeneral Set-up: Analyzing theStructure andRepresentations ofanArbitrarySemisimple LieAlgebra 197 $14.1: Analyzing SimpleLieAlgebrasinGeneral 197 §142: About theKilling Form 206 15,sl.C and s1€ ait $154: Analyzing sC a§15.2:Representations ofsi4Cands1,C 217§153: Weyl's Construction andTensor Products 22$154:SomeMoreGeometry 278155: Representations ofGLC 2 16.Symplectic LieAlgebras 238 §16.1:TheStructureofSp,,€andsps,C 238 §162: Representations ofsp 24 17.speCandsp,,€ 253 $17.4: Representations ofsp4C 253 $17.2: Representations ofspa,€ inGeneral 259 §17.3: Weyl's Construction forSymplectic Groups 262 18.Orthogonal LieAlgebras 267 $18.1 SO,Cand#0,€ 267 $18.2: Representations ofs05€, s04C, andsoy m3 19,s06C,£0,€,and50,€ 282 $19.1: Representations ofso 292§19.2:Representations ofthe Even Orthogonal Algebras 286 4193: Representationsofsoy 22 §19-4: Representations ofthe Odd Orthogonal Algebras 294 $19.5: Weyl's Construction forOrthogonal Groups 296 aiv Contents 20.Spin Representations of90, 299 §20.1:CliffordAlgebrasandSpinRepresentations of90, 9§202:TheSpinGroupsSping€andSpin, 307§203: Sping€ and Triality 312 Part IV: LieTheory 317 21.TheClassification ofComplex Simple LieAlgebras 319 $21.1: Dynkin Diagrams Associated toSemisimple LieAlgebras a9 $21.2: Classifying Dynkin Diagrams 32s §21.3 Recovering aLieAlgebra from ItsDynkin Diagram 330 22.g,andOther Exceptional LieAlgebras 339 £221:Construction ofgyfromItsDynkinDiagram 339§222: Verifying That g3isaLieAlgebra 346 $223: Representations of@s 350£2244:AlgebraicConstructions oftheExceptional LieAlgebras 39) 23,Complex LieGroups; Characters 366 §234: Representations ofComplex Simple Groups 36 §232: Representation Rings and Characters 375 §232: Homogeneous Spaces 382 8234: Brohat Decompositions 395 24,Weyl Character Formula 399 $241: TheWeyl Character Formula 399 £242: Applications toClassical LieAlgebras andGroups 403 25. More Character Formulas 41s §25.1: Freudenthal's Multiplicity Formula as §252: Proofof(WCF),theKostantMultiplicityFormula 419 §253: Tensor Products and Restrictions toSubgroups a4 26,Real LieAlgebras and LieGroups 430 §26.1: Classification ofReal Simple LieAlgebras andGroups 430 §26.2: Second ProofofWeyl'sCharacterFormula 440 £263:Real,Complex,andQuaternionie Representations aa ‘Appendices 451 ‘A.OnSymmetric Functions 453 $A: Basic Symmetric Polynomials and Relations among Them 453 §A2: Proofs ofthe Determinantal Mdentiies 402 §A3: Other Determinantal Kdentites 465 Contents, w B.OnMultilinear Algebra an $0.1: Tensor Products an $82: Exterior and Symmetric Powers an $0.3: Duals and Contractions 475 C.OnSemisimplicity 478 $C: The Killing Form and Cartan’s Criterion a8 §C2: Complete Reducibilty andtheJordan Decomposition a§C3:OnDerivations 33 D.Cartan Subelgebras 487 §D.l:TheExistenceofCartanSubalgebras 47{§D.2:OntheStructureofSemisimple LieAlgebras 489§D.3: TheConjugncy ofCartan Subalgebras 9 §D.4: OntheWey! Group 493 E, Ado’s and Levi's Theorems 499 §E.t: Levis Theorem 9 §E.2: Ado's Theorem eo F.Invariant Theory fortheClassical Groups 504 §F-A: The Polynomial Invariants soa §F.2: Applications toSymplectic and Orthogonal Groups su §F.3: ProofofCapeli’sIdentity sia Hints, Answers, and References 516 Bibliography 536 Index ofSymbols 543 Index S47 PART I FINITE GROUPS Given that over three-quarters ofthis book isdevoted totherepresentation theory ofLiegroups and Liealgebras, why have adiscussion oftherepresen- {ations offinite groups atall?There arecertainly valid reasons from alogicalpointofview:manyoftheideas,concepts,andconstructions wewillintroduce here willbeapplied inthestudy ofLiegroups andalgebras, The realreasonforus,however,isdidactic,aswewillnowtrytoexplain.Representation theoryisverymucha20th-century subject,inthefollowing sense. Inthe19th century, when groups were dealt with they were generallyunderstood tobesubsetsofthepermutations ofaset,oroftheautomor-phismsGL(Vofavectorspace¥,closedundercomposition andinverse.Only inthe20th century was thenotion ofanabstract group given, making it possible tomake adistinction between properties oftheabstract group and properties ofthe particular realization asasubgroupofapermutation group orGL(V). Togive ananalogy, inthe19th century amanifold wasalways a subsetoff*;onlyinthe20thcenturydidthenotionofanabstractRiemannianmanifold become common. Inbothcases,theintroduction oftheabstractobjectmade@fundamentaldifference tothesubject. Indifferential geometry, onecould inake acrucial distinction between the intrinsicandextrinsicgeometryofthemanifold:which propertieswere invariants ofthe metric on the manifold and which were properties oftheparticular embedding inR*.Questions ofexistence ornon- ‘existence, forexample, could bebroken upinto twoparts: didtheabstract manifold exist, andcould itbeembedded, Similarly, what would have been called inthe19th century simply “group theory” isnow factored into (wo parts. First, there isthestudy ofthestructure ofabstract groups (e, the classification ofsimple groups). Second isthecompanion question: given a group G,how canwedescribe alltheways inwhich Gmay beembedded in 2 1.FiniteGroups (ormappedto)alineargroupGL(VY?.This,ofcourseisthesubjectmatterofrepresentation theory. Given thispoint ofview, itmakes sense when firstintroducing representa-tiontheorytodosoinacontextwherethenatureofthegroupsGinquestionisitselfsimple,andrelativelywellunderstood. Iislargelyforthisreasonthatwwearestartingoffwiththerepresentation theoryoffinitegroups:forthosereaders who arenot already familiar with themotivations and goals of representation theory,itseemedbettertoestablishthosefirstinasettingwhere thestructure ofthegroups was not itself anissue. When weanalyze, for‘example,therepresentations ofthe symmetric and alternating groups on3,4, and 5letters, itcanbeexpected that thereader isalready familiar with the ‘groups andcanfocus onthebasic concepts ofrepresentation theory being introduced. ‘Wewillspendthefirssixlecturesonthecaseoffinitegroups.Manyofthe techniques developed forfinite groups will carry over toLiegroups; indeed,‘ourchoiceoftopicsisinpartguidedbythis.Forexample,wespendquiteabitoftimeonthesymmetric group;thisispartlyforitsowninterest,butalsopartly because what welearn here gives oneway tostudy representations of thegeneral linear group and itssubgroups. There areother topics, such asthe alternating group ®,,andthegroups SL;(F,) andGL,(F,) thatarestudied purely fortheir own interest and donotappear later. (Ingeneral, forthose teaders primarily concerned with Lietheory, wehave tried toindicate intheintroductory notestoeachlecturewhichideaswillbeusefulinthesucceedingparts ofthisbook.) Nonetheless, thisisbynomeans acomprehensive treat- ‘ment oftherepresentation theory offinite groups; many important topics, such astheArtin and Brauer theorems and thewhole subject ofmodular fepresentations, areomitted. LECTURE 1 Representations ofFinite Groups Jnthislecturewegivethebasicdefnitionsofrepresentation theory,andprovetwoofthebasicresultsshowingthateveryrepresentation i(unique)directsumofirreduc-ibleones. Wework outasexamples thecase ofabelian groups, and thesimplestnonabelian group,thesymmetric groupon3letters.Inthelattercasewegiveananalysis that willturn outnottobeuseful forthestudy offinite groups, butwhose ‘main deaiscentraltothestudyoftherepresentations ofLiegroups, 1.1: Definitions §1.2: Complete reducibility;Schurs lemma 1.3: Examples: Abelian groups; S, §1.1. Definitions {Arepresentation ofafinitegroupGonafnite-dimensional complexveetorspace Visahomomorphism p:G+GL(V) ofGtothegroup ofautomor- phismsofV;wesaythatsuchamapgivesVthestructure ofaG-module. When there islittle ambiguity about themap p(and, we're afraid, even sometimes‘whenthereis)wesometimes callVitselfarepresentation ofG;inthisveinwe willoften suppress thesymbol pandwriteg-vorguforp(g){v).Thedimension ofVissometimes called thedegree ofp.‘Amap@betweentworepresentations VandWofGisavectorspacemap 9:V+ Wsuch that vow vow ‘4 1.Representations ofFiniteGroups commutes forevery g€G.(We willcallthisaG-linear map when wewant to distinguish itfrom anarbitrary linear map between thevector spaces Vand W)Wecanthen define Ker g,Im9,and Coker @,whicharealsoG-modules. ‘Asubrepresentationofrepresentation VisavectorsubspaceWofVwhich isinvariantunderG.Arepresentation Viscalledirreducible ifthereisnoproper nonzero invariant subspace WofV.ITVandWarerepresentations, thedirectsumV&WandthetensorproductV@Warealsorepresentations, thelattervia 9(°@ )=go@aw. Forarepresentation ¥,thenthtensorpowerV®*isagainarepresentation of,Gbythisrule, andtheexterior powers /°(V) andsymmetric powers Sym"(V)aresubrepresentations' ofit.ThedualV*=Hom(V,€)ofVisalsoarepresentation, though notinthemost obvious way: wewant thetworepresenta- tions ofGtorespect thenatural pairing (denoted ¢,))between V*and ¥,sothatifp:G+GL(V)isarepresentation andp*:G+GL(V*)isthedual,weshould have <p*(a)(0*), a(a)(0)> =<v*, > forall g€G, ve¥,and v*€V*.This inturn forces ustodefine thedual representation by 9a) ="91°": V4+¥* forallg€6. Exercise L.1.Verify that with this definition ofp*,therelation above is satisfied Havingdefinedthedualofarepresentation andthetensorproductoftwo representations,itislikewisethecasethatifVandWarerepresentations, then Hom(¥, W)isalso arepresentation, viatheidentification Hom(¥, W)= V*@W.Unraveling this,ifweview anelement ofHom(¥, W)as.linear map@fromVtoW,wehave (a0)(0) =ao(a"'®) forallve¥.Inother words, thedefinition issuchthatthediagram vtow vw commutes. Note that thedual representation is,inturn, aspecial case ofthis: +Formoreonexteriorandsymmetricpowers.inludingdescriptionsasquotientspacesoftentor powers,eeAppendix 51.2. Complete Redvcibility; Schur's Lemma s whenW=Cisthetrivialrepresentation, ic,gw=wforallw€C,thismakesV*intoaG-module, withg9(0)=og'0),ic,a9=(ao. Exercise{.2.VerifythatingeneralthevectorspaceofG-linear maps between tworepresentations Vand WofGisjustthesubspace Hom(V, W)® of elements ofHom(¥, W)fixed under theaction ofG.This subspace isoften denoted Homa(¥, Wehave, ineffec, taken theidentification Hom(¥, W)=V*@Wasthe definition oftherepresentation Hom(¥, W).More generally, theusual iden- tities forvector spaces arealso true forrepresentations, cg, VEUGM=VENEVEM, NVOM= @NVONW, A=, and soon, Exercise .3°. Letp:G+ GL(V) beanyrepresentation ofthefinite group G onann-dimensional vector space Vand suppose that foranyg€G,the determinant ofp(g) is1.Show that thespaces MV andA'-*V* areisomorphicasrepresentations ofG. IfXisanyfinite setand Gacts ontheleftonX,ie,G+Aut(X) is homomorphism tothepermutation group ofX,there isanassociated per-‘mutationrepresentation: letVbethevectorspacewithbasis(e,:x€X},andletGactonVby OLMee =Fae. ‘The regular representation, denoted RgoFR,corresponds totheleftaction of Gonitself. Alternatively, Risthespace ofcomplex-valued functions onG, where anelement g€Gactsonafunction «by(g2)(h) =(gh. Exercise 1.4*. (a)Verify thatthese (wodescriptions ofRagree, byidentifying theelement e,with thecharacteristic function which takes thevalue fonx,O.onotherelementsofG.(b)Thespace offunctions onGcanalsobemade into aG-module bythe rule (g2)(h) =a(fig). Show that thisisan isomorphic representation. §1.2. Complete Reducibility; Schur’s Lemma {Asinany study, before webegin our attempt toclassify therepresentations ofafinitegroupGinearnestweshouldtrytosimplifylifebyrestricting oursearchsomewhat, Specifically, wehaveseenthatrepresentations ofGcanbe 6 1.Representations ofFiniteGroups built upoutofother representations bylinear algebraic operations, most simply bytaking thedirect sum. Weshould focus, then, onrepresentations that are“atomic” with respect tothisoperation,ie.thatcannotbeexpressed asadirect sum ofothers; theusual term forsuch arepresentation isinde- composable. Happily, thesituation isasnice asitcould possibly be:arepre-sentationisatomicinthissenseifandonlyitisirreducible (ie.containsnoproper subrepresentations), and every representation isthedirect sum of ireducibles, inasuitable sense uniquely so.The keytoalthisis Proposition 1.5.IfWisasubrepresentation ofarepresentation Vofafinite group G,thenthere isacomplementary incariant subspace W"ofV,sothat v-wew. Proor. There aretwo ways ofdoing this. One canintroduce a(positive definite) Hermitian inner product HonVwhich ispreserved byeach g€G(ie,suchthatH(g0,gw)=H(o,w)forall»,w€Vandg€G).Indeed,ifHyis.any Hermitian product on¥,onegets such anHbyaveraging over G: U(0,»)=Y.Hoge, aw). ‘Then theperpendicular subspace W!iscomplementary toWinV.Alterna- Lively (butsimilarly), wecansimply choose anarbitrary subspace Ucomple- mentary toW,letxo:V+Wbetheprojection given bythedirect sum decomposition V=W@U,andaverage themap x»over G:that is,take Ho)=¥,altel to This willthen beaG-linear map from Vonto W,which ismultiplication by IG\onW;its kernel will, therefore, beasubspace ofVinvariant under Gand complementary toW. o Corollary 1.6.Anyrepresentation isadirectsumofirreducible representations. This property iscalled complete reducibilty, orsemisiplicity. Wewillsee that,forcontinuous representations thecircle$?,oranycompact group, hasthisproperty;integration overthegroup(withrespecttoaninvariantmeasure‘onthegroup) plays theroleofaveraging intheabove proof. The (additive) group Rdoes nothave thisproperty: therepresentation a(t 4 O14 leaves thexaxisfixed, butthere isnocomplementary subspace. Wewillsee other Liegroups such asSL,(C) thataresemisimple inthissense. Note also that thisargument would failifthe vector space Vwasover afield offinitecharacteristicsinceit mightthenbethecasethatx(2)=OforveW.Thefailure $1.2. Complete Redueibiity; Schur’s Lemma 7 ofcompletereducibility isoneofthethingsthatmakesthesubjectofmodularrepresentations, orrepresentations onvectorspacesoverfinitefields,sotricky. ‘Theextenttowhichthedecomposition ofanarbitraryrepresentation into adirect sum ofirreducible ones isunique isone oftheconsequences ofthe following: Schur's Lemma 1.7.IfVand Wareirreducible representations ofGand @:V+ Wisa G-module homomorphism, then (1)Either¢isanisomorphism, or=0.Q)IfV=Wytheng=2-1forsomeheC,theidentity. Proor.ThefirstclaimfollowsfromthefactthatKer@andImareinvariantsubspaces. Forthesecond, since €isalgebraically closed, must have an cigenvalue 2,ie,forsome 1€€, ¢~AIhasanonzero kernel. By(1),then,wemusthave@~Al=0,50=Al. a Wecansummarize whatwehaveshownsofarin Proposition 1.8.For anyrepresentation Vofafinite group G,there isa decomposition V=Ve@-@ Ko, where theV,aredistinct irreducible representations. The decomposition ofV intoadirect sumofthekfactors isunique, asarethe¥,that occur andtheir ‘multiplicities a, Proor. Itfollows from Schur’s lemma that ifWisanother representation of G,withadecomposition W=@W,°, ando:V+Wisamapofrepresen- tations, thengmust map thefactor V;%"intothatfactor W;®™ forwhich W,=Vi;when applied totheidentity map ofVtoV,thestated uniquenessfollows. a Inthenextlecture wewillgiveaformula forthe projection ofVonto Vi".‘Thedecomposition oftheithsummandintoadirectsumofa,copiesofVis notunique ifa,>1,however. Occasionally thedecomposition iswritten, Va aKO-@ahk=ahi+tah, (19) especially when oneisconcerned only about theisomorphisin classes and multiplicities ofthe‘Onemorefactthatwillbeestablished inthefollowinglectureisthatafinite ‘roup Gadmits only finitely many irreducible representations V,uptoiso-morphism (infact,wewillsayhowmany).This,then,istheframework oftheclassification ofallrepresentations ofG:bytheabove,oncewehavedescribed 8 1.Representations ofFinite Groups theiereducible representations ofG,wewillbeable todescribe anarbitrary representation asalinear combination ofthese. Our firstgoal, inanalyzingtherepresentations ofanygroup,willthereforebe: (Describe alltheirreducible representations ofG. ‘Oncewehavedonethistheretemainstheproblemofcarrying outinpracticethedescription ofagivenrepresentation intheseterms.Thus,oursecondgoalwill be: (ii)Find techniques forgiving thedirect sum decomposition (1.9), and inparticulardetermining themultiplicities a,ofanarbitraryrepresentation V. Finally, itithecase thattherepresentations wewillmost often beconcerned with arethose arising from simpler ones bythesort oflinear- ormultiinear- algebraic operations described above, Wewould like, therefore, tobeable to <escribe, inthe terms above, therepresentation wegetwhen weperform these ‘operations onaknown representation. This isknown generally as (ii) Plethysm: Describe thedecompositions, with multiplicities, ofrepresen- tations derived from aqiven representation ¥,such asV@FV, V*,MV), Sym'(V), andAi(A'V), Note thatifVdecomposes intoasumoftworepresen- tations, these representations decompose accordingly;eg,if V=U@W,then NV=@Nu@Nw, s0itisenough towork outthisplethysm forirreducible representations. Similarly, ifVandWaretwoirreducible representations, wewant todecom- pose V@W;thisisusually known astheClebsch-Gordon problem. §1.3. Examples: Abelian Groups; S3 One obvious place tolook forexamples iswith abelian groups. Itdoes not take ong, however, todeal with thiscase. Basically, wemay observe ingeneral that ifVisarepresentation ofthefinite group G,abelian ornot,each g€G givesamapp(g):V-+V;butthismapisnotgenerally aG-module homomor- phism: forgeneral h€Gwewillhave ai(on) ¥H(gto)- Indeed,p(g):V-+¥willbeG-linearforeverypif(andonlyif)gisinthecenter, 2(G)ofG.InparticularifGisabelian,andVisanirreducible representation, then bySchur's lemma every element g€Gacts onVbyascalar multiple of theidentity. Every subspace ofVisthus invariant; sothat Vmust beone«dimensional, Theirreducible representations ofanabeliangroupGarethus‘simplyelementsofthedualgroup,thatis,homomorphisms G40 §1.3. Examples: Abelian Groups; S 9 Weconsider next thesimplest nonabelian group, G=G4.Tobegin with, wwehave (aswith anysymmetric group) twoone-dimensional representations: wehave thetrivial representation, which wewilldenote U,andthealternating representation U,defined bysetting 0=san(g)o forg€G,v€ C.Next, since Gcomes tousasapermutation group, wehave ‘@natural permutation representation, inwhich GactsonC*bypermutingthecoordinates. Explicitly,if{e),e3,¢s) isthestandard basis,theng-€,=ey.or,equivalently, (24523523)=Ga-sayZarvanFeton) ‘This representation, likeanypermutation representation, isnotirreducible: thelinespanned bythesum (1,1, 1)ofthebasis vectors isinvariant, with ‘complementary subspace Vm((2y,2y:25)C82,+25425=O) ‘Thistwo-dimensional representation ¥iseaslyseentobeirreducible;wecall itthestandard representation ofSy. Letusnow turn totheproblem ofdescribing anarbitrary representation ofS.Wewillseein thenext lecture awonderful tool fordoing this, called‘charactertheory;but,asinefficientasthismaybe,wewouldlikeheretoadoptamore adhocapproach. This hassome virtues asadidactic technique inthe Present context (admittedly dubious ones, consisting mainly ofmaking thepointthatthereareotherandfarworsewaysofdoingthingsthancharactertheory). The real reason wearedoing itis that itwill serve tointroduce anideathat,whilesuperfluous foranalyzing therepresentationsoffinitegroups ingeneral, willprove tobetheKeytounderstanding representations ofLie groups. ‘Theideaisverysimple one:since wehave justseen thattherepresentation theory ofafiniteabeliangroupisvirtuallytrivial,wewillstartouranalysis ‘ofanarbitrary representation Wof&,bylooking just attheaction ofthe abelian subgroup Wl,=Z/3¢ &,onW.This yields avery simple decom- position: ifwetake ttobeanygenerator ofWl(that is,anythree-cycle), the space Wisspanned byeigenvectors», fortheaction ofx,whose eigenvaluesareofcourseallpowersofacuberootofunity«=e?*".Thus, w=OY where Vin Cy and r= 0% Next,weaskhowtheremaining elementsofS,actonIVintermsofthis decomposition. Toseehow thisgoes,letobeanytransposition,sothat+and «together generate S,,withtherelation ota=x?Wewant toknow where ‘sends aneigenvector forthe action ofr,saywitheigenvalue «;toanswer 10 1.Representations ofFinite Groups this,welookathowractsono(s).Weusethebasicrelationabovetowrite Ho(0)) =o(¢°(0) =o(w"-0) =0o(0) ‘The conclusion, then,isthativisaneigenvector forxwitheigenvalueco,then (0)isagain aneigenvector fort, with eigenvalue w. Exercise 1.10, Verify thatwith o=(12), r=(123),thestandardrepresentation has abasisa=(«,1,0"),=(Ly,0"),with =o, f=o'f, of, ofa. Suppose now that westart with such aneigenvector vforr.Ifthe eigenvalue ofviswo¥1,then o(0)isaneigenvector witheigenvalue «w™¥«and sois. independent ofv;andvando(o)together span atwo-dimensional subspace V’ofWinvariant under 5.Infact, Vis isomorphic tothestandard repre- sentation, which follows from Exercise 1.10. If,ontheother hand, theeigen- value ofvis1,then o(0)may ormay notbeindependent ofv.Ifit isnot,then spans aone-dimensional subrepresentation ofW,isomorphic tothetrivialrepresentation if(0)=oandtothealternatingrepresentation ifo(0)=—v. Mfa(o)andvareindependent, thenv+o(o)andv—o(o)spanone-dimensional representationsofWisomorphictothetrivialandalternatingrepresentations, respectively Wehave thus accomplished thefirst two ofthegoals wehave setforourselvesaboveintheeaseofthegroupG=Sy.First,weseefromtheabovethattheonlythreeirreducible representations ofS,arethetrivial,alternating,andstandard representations U,U'and V.Moreover, foranarbitrary representationWofS,wecanwrite Wa U™@UM@Ve; andwehave awaytodetermine themultiplicities a,b,andc:cforexample, isthenumber ofindependent eigenvectors for¢with eigenvalue «,whereasa+cisthemultiplicity oflasaneigenvalue ofa,andb+cisthemultiplicity, of—1asaneigenvalue ofo.Infact, thisapproach gives usaswell theanswer toourthird problem,findingthedecomposition ofthesymmetric, alternating, ortensorpowersofagiven representation W,since ifweknow theeigenvalues oftonsucha representation,weknowtheeigenvalues oftonthevarioustensorpowersof W.Forexample, wecanusethismethod todecompose V@¥,where Visthestandardtwo-dimensional representation. ForV@Visspannedbythevectors «Ba, «Bf, PB, and \®P; these areeigenvectors fortwithcigenvalues 7%,1,1,andc,respectively, andainterchanges @@«with1f,andx@[email protected]«@xandf@fspanasubrepresentation $1.3. Examples: Abelian Groups; Gy " isomorphic to¥,2@f+P@a spans atrivial representation U,and &® ff @aspans U’,s0 VeVvsv@u'ey. Exercise 1.11. Usethisapproach tofind thedecomposition oftherepresen- tations Sym?V andSym? Exercise1.12.(a)Decompose theregularrepresentation RofG5,(b)Show that Sym**SV isisomorphic toSym'V@R,andcompute Sym*V for allk. Exercise 1.13*. Show that Sym?(Sym?V) =Sym?(Sym?¥). IsSym"(Sym*V)isomorphic toSym"(Sym*V)? ‘Aswehaveindicated, theideaofstudyingarepresentation VofagroupG byfirstrestricting theactiontoanabeliansubgroup, gettingadecomposition ofVinto one-dimensional invariant subspaces, and then asking how the remaining generators ofthe group actonthese subspaces, does notwork well forfinite Gingeneral; forone thing, there willnotingeneral beaconvenient abelian subgroup touse. This idea will turn out, however, tobethekey to understanding therepresentations ofLiegroups, with atorus subgroup playing theroleofthecyclic subgroup inthisexample. Exercise 1.14*, LetVbeanirreducible representation ofthefinite group G. Show that, uptoscalars, there isaunique Hermitian inner product on¥ preserved byG. LECTURE 2 Characters “Thislecturecontaintheheartofourtreatmentoftherepresentation theoryoffinite‘groups:thedefinition in§2.1ofthecharacter ofarepresentation, andthemaintheorem (provedintwostepsin§2.2and§2.4)thatthecharacters oftheirreducible representa- tionsformanorthonormal bassforthespaceofclasfunctionsonG.Theewillbe ‘oreexamplesandmoreconstructions inthefollowinglectures,butthisiswhatyou teed 10know. $21: Characters$22Therstprojectionformulaandisconsequences$23 Examples: S,and Mt, §24: More projection formulas; more consequences §2.1. Characters Asweindicated inthepreceding section, there isaremarkably effective tool forunderstanding the representations ofafinite group G,called character theory. This isinsome ways motivated bytheexample worked out inthelastsection where wesawthat arepresentation ofSywasdeterminedbyknowing theeigenvalues oftheactionoftheelements tando€Ss.Forageneral group G,itisnotclear what subgroups and/or elements should play therole ofWs, t,and 0;buttheexample certainly suggests that knowing alltheeigenvalues ofeach element ofGshould sullice todescribe the representation. ‘Ofcourse,specifying alltheeigenvalues oftheactionofeachelementofG issomewhat unwieldy, butfortunately itis redundant aswell. For example,ifweknowtheeigenvalues {2,}ofanelementg€G,thenofcourseweknowtheeigenvalues {4*}ofg*foreachkaswell.Wecanthususethisredundancy G21. Characters b tosimplify thedatawehavetospecify.Thekeyobservation hereisitisenoughtogive,forexample,justthesumoftheeigenvalues ofeach element ofG,since knowing thesumsJ:Afofthekthpowersoftheeigenvalues ofagivenelement@€Gisequivalent foknowing theeigenvalues {2,}ofgthemselves.Thisthen suggests thefollowing: Definition. IfVisarepresentation ofG,itscharacter xyisthecomplex-valued function onthegroup defined by alg) =Tale) thetraceofgon¥. Inparticular, wehave Aelligh™) =ro(9 ‘80thatxyisconstant ontheconjugacy classesofG;suchafunctioniscalled aclass function. Note that yy(1) =dim V. Proposition 2.1.LetVandWberepresentations ofG.Then Xrew =e tt ven =kyr a= Fyand xnry(9) =Meo)? ~rola) Proor. Wecompute thevalues ofthese characters onafixed element g€G. Fortheaction ofg,Vhaseigenvalues {2,}andWhaseigenvalues {y1,}. Then{4+sy}and{4,°1))areeigenvalues forV@WandV@W,fromwhichthefirsttwoformulas follow.Similarly (4;=2,}aretheeigenvalues forgonV*,since alleigenvalues arenthroots ofunity, with ntheorder ofg.Finally, {Adjli <j}aretheeigenvalues forgonA?¥,and AP -D4,ae andsince g?haseigenvalues {42}, thelastformula follows. a Exercise 2.2.ForSym?V, verify that Aynov(@) =Hava)? +x09) Note thatthisiscompatible with thedecomposition V@V =Sym’V OAV. Exercise 2.3%. Compute thecharacters ofSym'V andA‘V. Exercise 24°. Show that ifweknow thecharacter zyofarepresentation V, then weknow theeigenvalues ofeach element gofGin thesense that we 4 2.Characters knowthecoefficients ofthecharacteristic polynomial ofg:V+V.Carrythis‘outexplicitly forelementsg€Goforders2,3,and4,andforarepresentation ofGonavectorspaceofdimension 2,3,or4. Exercise 25.(The original fixed-point formula). Visthepermutation repre- sentation associated totheaction ofagroup Gonafinite setX,show that(9)isthenumberofelementsofXfixedbyg. ‘Aswehavesaid,thecharacter ofarepresentation ofagroupGisreallya functiononthesetofconjugacy classesinG.Thissuggestsexpressing thebasicinformation abouttheirreducible representations ofagroupGintheformof1charactertable.Thisisatablewiththeconjugacy classes[]ofGlistedacross thetop, usually given byarepresentative g,with(forreasonsthatwill becomeapparentlater)thenumberofelementsineachconjugacyclassover it;theirreducible representations ¥ofGlisted ontheleft;and, intheappro-priatebox,thevalueofthe character xyontheconjugacy class [4 Example 26.Wecompute thecharacter table of©.This iseasy: tobeginwith,thetrivialrepresentation takesthevalues(I,1,1)onthethreeconjugacyclasses [1], [(12)}, and [(123)}, whereas thealternating representation has values (1,~1, 1).Toseethecharacter ofthestandard representation, notethatthepermutation representation decomposes: €?=U@V;sincethecharacter ofthepermutation representation has,byExercise 2.5,thevalues 1,0), wehave zy=ze—tu=Gs1,0) (1,1, 1)=(2,0, 1). Insum, then, thecharacter table ofS,is 13002 &|1ayay wivaty [I 1alternating U'|1-1standad¥ 120 00 =t ThisgivesusanothersolutionofthebasicproblemposedinLecture1:ifWis any representation ofS,and wedecompose Winto irreducible repre- sentations W=U®*® U'®® V%, then tw=ary+bry: +Cxy. Inparticu- lar,since thefunctions xy,xy-and xyare independent, weseethat Wis determined uptoisomorphism byitscharacter Yy- Consider, forexample, [email protected] (zy)?, which hasvalues 4,0, and |onthethree conjugacy classes. Since V®U®U’hasthesame char- acter, this implies that V@Vdecomposes into V@U@U',aswehave seen directly. Similarly, V@U'hasvalues 2,0,and —1,so¥@ U"x¥. $22. TheFirst Projection Formula andItsConsequences Is Exercise 2.7*, Find thedecomposition oftherepresentation V®*using char- acter theory. Characters willbe similarly useful forlarger groups, although its rare to {findsimple closed formulas fordecomposing tensor products. §2.2. The First Projection Formula and ItsConsequences Inthelastlecture, weasked (among other things) foraway oflocating explicitly thedirectsumfactorsinthedecomposition ofarepresentation into irreducible ones. Inthissection wewillstart bygiving anexplicit formula fortheprojectionofanirreducible representation ontothedirectsumofthetrivialfactors inthisdecomposition; asitwill turn out, thisformula alone has tremendous consequences. Tostart, forany representation Vofagroup G,weset Vo={veV:go=0 YgeG}. Weaskforaway offinding V°explicitly. The idea behind oursolution to thisisalready implicit intheprevious lecture. Weobserved there that foranyrepresentation VofGandanyg€G,theendomorphism g:V-»Vis,ingeneral, notaG-module homomorphism. Ontheother hand, ifwetake the average ofallthese endomorphisms, that is,weset 1 -t End(v)ogi dane Baar thentheendomorphism ¢willbeG-linearsinceSig=hglt'.Infact,wehave Proposition 2.8.Themap@isaprojection ofVontoV°. Proor.First,suppose v=(Ww)=(1/|G|)Dgw.Then,foranyheG, 1 1 w= Shaw=1-5 aw,0GEhw=gjZom ‘sotheimage ofgiscontained inV°.Conversely, ifveV%,then p(v)=(NG)Lv=0,80V7=Imighand 929=9. fal Wethus have away offinding explicitly thedirect sum ofthetrivial subrepresentations ofagiven representation, although theformula can be hard touseifitdoes notsimplify. Ifwejust want toknow thenumber mof copies ofthetrivial representation appearing inthedecomposition ofV,we can dothis numerically, since this number will bejust the trace ofthe 16 2Characters projection 9.Wehave sm=dim VO=Teace() 1 1=DLTrace(g) =— v( 29)iideMO=GF ? Inparticular, weobserve that foranirreducible representation Vother than thetrivial one, thesum over allg€Gofthevaluesofthecharacteryyiszero. Wecan domuch more with thisidea, however. The keyistouseExercise 1.2:ifVandWarerepresentations ofG,thenwithHom(V, W),therepresenta- tion defined inLecture 1,wehave Hoin(V, W)®=(G-module homomorphisms froin VtoW). ICVisirreducible thenbySchur'slemmadimHom(¥,W)?isthemultiplicityofVinW;similarly, ifWisirreducible, dimHom(V, W)?isthemultiplicity ofWinV,andinthecasewherebothVandWareirreducible, wehave " 1iftvowdimHom(V,W)={aw Butnowthecharacter yon. ofthe representation Hom(V, W)=V*@W isgiven by Aino (9) =HAG) Lor). Wecan now apply formula (2.9) inthiscase toobtain thestriking — 1itvewgidMemo ={tvew. 2.10) To-express this lt Ctoa(G) =lass functions onG) anddefine anHermitian inner product onCq,,,,(G) by — (ap) =b ai(eA)=ig3,OM eu Formula (2.10) then amounts to ‘Theorem 2.12, Interms ofthisinner product, thecharacters oftheirreductble representations ofGareorthonormal. For example, theorthonormality ofthethre irreducible representations ‘ofG,canbereadfromitscharacter tableinExample2.6.Thenumbersover each conjugacy class tellhow many times tocount entries inthat column. Corollary 2.13. Themumber ofireducibe representations ofGis lessthan or equal 10thenumberof con)ugacy classes. $22.TheFirstProjectionFormulaandItsConsequences 0 Wewill soon show that there arenononzero class functions orthogonal tothecharacters, sothat equality holds inCorollary 2.13. Corollary 2.14. Anyrepresentation isdetermined byitscharacter. IndeedifV=Vi"@-~-@%,withthe¥,distinctirreducible characters,then xy=Dayty, andthezy,arelinearly independent, Corollary2.15.Arepresentation Visirreducible ifandonlyif(ysty)=1. Infact, ifVxVi" @+--@ HO% asabove,then(xv.Xr)=Sa ‘The multiplicities a,can becalculated via Corollary 2.16.Themultiplicity aof¥,inVistheinnerproduct ofxywithx.jie,a)=(tvsXv) Weobtain some further corollaries byapplying allthistotheregularrepresentation RofG.First,byExercise2.5weknowthecharacterofR;itis simply _{0ifowe ule)fritg=e. Thus, weseefistofall that Risnotirreducible ifG#{e}.Infact, ifweset R=@YV,%, with V,distinct irreducibles, then 1 , 41™Otoa)=jgqptederIGI =dimYi ean Corollary 2.18. Any irreducible representation VofGappears intheregular representation dim Vtimes. Inparticular, thisprovesagainthatthereareonlyfinitelymanyirreduciblerepresentations. Asanumerical consequence ofthis wehave theformula 1G]=dim(R) =¥dim(viy 19) Also, applying thistothevalue ofthe character oftheregular representation ‘onan clement g€Gother than theidentity, wehave 0=Lidim ¥)-zv(9) ig#e. (2.20) These twoformulas amount totheFourier inversion formula forfinite groups, fExample 3.32. Forexample, ifllbutoneofthecharacters isknown, they sive aformula fortheunknown character. Exercise 221. The orthogonality oftherows ofthecharacter table isequiv- alent toanorthogonality forthecolumns (assuming thefactthatthere areas 1" 2.Characters manyrowsa8columns).Writtenout,thissays:()ForgeG, — _1GlExot =T5 Where thesum isover allirreducible characters, and c(g) isthenumber of elements intheconjugacy class ofg. Gi)IFgandhareelementsofGthatarenotconjugate,then Lxio)xlh) =0. Notethatforg=¢thesereduceto(2.19)and(2.20). §2.3. Examples: S,and %, Toseehowtheanalysisofthecharacters ofagroupactuallygoesinpractice,wenow work outthecharacter table ofS,.Tostart, welisttheconjugacyclassesinS,andthenumberofelements ofS,ineach.Aswithanysymmetric group S;,theconjugacy classes correspond naturally tothepartitions ofd, that is,expressions ofdasasum ofpositive integers ay,a3,...,44,Where thecorrespondence associates tosuch apartition theconjugacy class ofa Permutation consisting ofdisjoint cycles oflength a),a3, ..,ay.Thus, inSy wwehave theclases oftheidentity element |(4= 1-41-41 +1),atrans- position such as(12), corresponding tothepartition 4=2+ 1+1;athree cycle (123) corresponding to4=3+1;afour-cycle (1234) (4=4);and theproductoftwodisjointtranspositions (12)(34)(4=2+2) Exercise 2.22. Show that thenumber ofelements ineach ofthese conjugacy classesis,respectively, 1,6,8,6,and3. AAsfortheirreducible representations ofSq, westart with thesame ones that wehad inthecase ofS,: thetrivial U,thealternating U',and thestandard representation V,i,thequotient ofthe permutation representation associated tothestandard action ofS,onasetoffour elements bythe Urivial subtepresentation. The character ofthetrivial representation onthe five conjugacy classes isofcourse (1,1, 1,1, 1),and that ofthealternating representation is(1,—1,1,—1,1).Tofind thecharacter ofthestandard {epresentation, weobservethatbyExercise25thecharacterofthe permuta- tionrepresentation onC*isxc«=(4,2,1,0,0)and,correspondingly, ty=tertv=G0,1,~2). Note that |p]=1,soVisirreducible, The character table sofarlooks like 23. Examples: 6,and M1, " 1 6 8 ‘ 3 St 0 (ry 34) (12904) wal |r1otfl 1atternating U|EE 1sanded 1300 10 at “1 Clearly, wearenotdone yet:since thesum ofthesquares ofthe dimensions ofthese three representations is1+1+9= LI,by(2.19) there must beadditional irreducible representations ofGg, thesquares ofwhose dimensionsaddupto24—11=13.SincetherearebyCorollary 2.13atmosttwoofthem,there must beexactly two, ofdimensions 2and 3.The latter ofthese iseasy tolocate: ifwejusttensor thestandard representation with thealternating, one U’,weatrive atareptesentation V’with character tv:=xy"te: = G,=1,0, 1,=. Wecanseethat this isirreducible either from itscharacter(since[zy-1=1)orfromthefactthatitisthetensorproductofanirreduciblerepresentation with aone-dimensional one; since itscharacteris notequal tothatofanyofthe firstthree, thismust beoneofthetwomissing ones. Asfortheremaining representation ofdegreetwo,wewillfornowsimplycallitW;wecandetermine itscharacter from theorthogonality relations (2.10). We obtain then thecomplete character table for4: 1 6 8 6 3 ef 0 ay) 02909 wivald [Tt q 1 stternating | to =E Lt 1standard¥ }30010 =I “1Veveu|3 -1 0 1 -1 Another 12 0 = ° 2 Exercise 2.23. Verify thelastrowofthistable from (2.10) or(2.20), Wenow getadividend: wecantake thecharacterofthemysteryrepresen- tation W,which wehave obtained from general character theory alone, and useittodescribe therepresentation Wexplicitly! The keyisthe2inthelast column foryw:thissays that theaction of(12)(34) onthetwo-dimensional vector space Mis aninvolution oftrace 2,and somust betheidentity. Thus, Wisreally arepresentation ofthequotient group! IEW isanormal subgroupof«groupGrepresentation p:G-+GL{V)itiionWifand only iitfactors vough thequotent G-GN-=-GUN. Representalons ofCIN canbeientied with epresentations ofGtha arrival onW. 2» 2Characters Si/{1,(12)(34), (13)(24),(14)(23)} &Ss, [One may seethisisomorphism byletting ©,actontheelements ofthe‘conjugacy classof(12)(34); equivalently, ifwerealize,asthegroupofrigid motions ofacube(seebelow),bylookingattheactionofS,onpairsof ‘opposite faces.] Wmust then bejustthestandard representation ofS,pulled back toG,viathisquotient. Example2.24.Aswesaidabove,thegroupofrigidmotionsofacubeisthesymmetric grouponfourletters;G,actsonthecubeviaitsactiononthefour Jong diagonals. Itfollows, ofcourse, that G,actsaswell onthesetoffaces. ofedges, ofvertices, ete; and toeach ofthese isassociated apermutationrepresentation ofG,.Wemaythusaskhowtheserepresentations decompose;‘WewilldoherethecaseofthefacesandIeavetheothersasexercises. We start, ofcourse, bydescribing thecharacter yofthepermutation representation associated tothefaces ofthecube. Rotation by180° about a linejoining themidpoints oftwoopposite edges isatransposition inS,and fixes nofaces, 0x(12) =0.Rotation by120° about along diagonal shows1(123)=0,Rotationby90°aboutalinejoiningthemidpointsoftwoopposite faces shows x(1234)=2,androtationby180°givesx((I2)(34))=2.Now (%x)=3,s0zsthesum ofthree distinct irreducible representations, From thetable, (x.20)=(xs%e-)=OsXw)=I,andtheinnerproductswiththe others arezero, sothis representation isU@V’@ W.Infact, thesums of ‘opposite faces span athree-dimensional subrepresentation which contains U (spanned bythesum ofallfaces), sothis representation isU@ W.Thedifferences ofoppositefacesthereforespan¥". Exercise 225*. Decompose thepermutation representation ofG,on(i)the vertices and(ii)theedges ofthecube. Exercise 2.26. The alternating group %,hasfour conjugacy classes. Three representations U,U',andU"come from therepresentations of Wa/{1,(12)(34),(139(24),(14)(23)} =2/3, s0there isone more irreducible representation Vofdimensidn 3.Compute thecharacter table, with «=e; boa 4 3 C7 ec vfi H 7 ult wt 1 wlio o 1 vis 0 0 “1 24. More Projection Formulas; More Consequences Fy Exercise 2.27. Consider therepresentations ofS,and their restrictions to%,. Which arestill irreducible when restricted, and which decompose? Which pairs ofnonisomorphic representations ofS,become isomorphic when restricted? Which representations of%,arise asrestrictions from ©,? §2.4. More Projection Formulas; More Consequences Inthissection, wecomplete theanalysis ofthecharacters ofthe irreducible representations ofageneralfinitegroupbegunin§2.2andgiveamoregeneral formula fortheprojection ofageneralrepresentation Vontothedirectsum ofthefactors inVisomorphic toagiven irreducible representation W.The main ideaforboth isageneralization ofthe“averaging”oftheendomorphisms 4:V~»Vusedin§2.2,thepointbeingthatinsteadofsimply averaging allthe gwecanaskthequestion: what linear combinations oftheendomorphisms g:V+VareG-linear endomorphisms? Theanswer isgiven by Proposition 2.28. Leta:G-+€ beanyfunction onthegroup G,andforany representation VofGset Gar=Lalg)-ai VV. Then18@homomorphism ofG-modulesforallVifandonlyifaisaclassSanction. Proor, Wesimply write outthecondition that ~,ybeG-linear, and theresult falls out: wehave Garth) = ag): g(he) =Lahgh)- hgh") (substituting hgh" forg) =MYathgh )-a(0)) =MEaa) a0) (fis aclase function) =Mg..r() Exercise 2.29*. Complete thisproof byshowing that conversely if@isnota classfunction,thenthereexistsarepresentation VofGforwhich@,failsto beG-linear. oO {Asanimmediate consequence ofthis proposition, wehave 2 2.Characters Proposition 230. Thenumber ofirreducible representations ofGtsequal tothenumberofconjugacy classesofG.Equivalently, theircharacters {xy}forman orthonormal basis forCunu(G)- Proor. Suppose a:G-» Cisaclassfunctionand(a,xy)=0forallirreducible representations V;wemust show thata=0.Consider theendomorphism oan=LegaVV asdefined above.BySchur'slemma,g,,,=A-Id;andifn=dimV,then ae!trmce(ouy) 1 =Lalo —=ea =o. Thus,»,.y=0,oFY.a(g)-g =0onanyrepresentation VofG;inparticular,thiswillbetruefortheregular representation V=R.ButinRtheelements {g€G}, thought ofaselements ofEnd(R), arelinearly independent. For ‘example, theelements {g(e)} areallindependent. Thus a(g)=0forallg,asrequired a ‘This proposition completes thedescription ofthecharacters ofafinite group ingeneral. Wewillseeinmore examples below how wecanusethis information tobuild upthecharacter table ofagiven group. Fornow, we mention another way ofexpressing thisproposition, viatherepresentation ring ofthegroup G. ‘Therepresentation ringR(G) ofgroup Giseasy todefine. First, asagroup wejusttake R(G) tobethefreeabelian group generated byall(isomorphism classes of)representations ofG,andmod outbythesubgroup generated byelementsofthe form ¥+W—(V @W),Equivalently, given thestatement of complete reducibility, wecanjusttakeallintegral linear combinationsa,"¥; oftheirreducible representations ¥,ofG;elements ofR(G)arecorrespondingly called virtual representations. The ting structureisthengivensimplybytensor product, defined onthegenerators ofR(G) andextended bylinearity. ‘Wecanexpressmostofwhatwehavelearnedsofaraboutrepresentations: ofafinitegroupGintheseterms.Tobegin,thecharacter definesamap 2R(G) +Caml) fromR(G)totheringofcomplex-valued functions onG;bythebasicformulasofProposition 2.1,this map isin facta ring homomorphism. The statement thatarepresentation indetermined byitscharacter thensaysthatyisinjective; $24, More Projection Formulas; More Consequences 2B theimagesofyarecalledvirtualcharacters andcorrespond thereby tovirtual representations. Finally, ourlast proposition amounts tothestatement that xcinduces anisomorphism Ae!RG) ®C-+ Coal) ‘The virtual characters ofGformalatticeA=Z*inCetses(G),inwhichthe actual characters sitasacone Ay N°c Z*We can thus think ofthe problem ofdescribing thecharacters ofGashavingtwoparts:first,wehave tofind A,and then thecone Ag<A(once weknow Ap,thecharactersofthe irreducible representations willbedetermined). Inthefollowing lecture wewillstatetheorems ofArtinandBrauercharacterizing A@QandA. ‘The argument forProposition 230 also suggests how toobtain amoregeneralprojectionformula.Explicitly,ifWisafxedirreduciblerepresentation,then forany representation V,look attheweighted sum 1=1.Yiwl@-aeEnd =Gi,twlairgeEndy, ByProposition 2.28, yisaG-modulehomomorphism. Hence,ifVisirreduc- ible, wehave w=A-Id, and 1 dmgap Tey -FEV [Gln@ wo)FanV[GHW wo 1 ={dimVivew 0 itvem. For arbitrary V, 173 vy=dim WGtelah: VV (231) istheprojection ofVontothefactorconsisting ofthesumofallcopiesofWappearing inV.Inother words, ifV=QV," then ty y= dim yD ila 2.32)Top (232) istheprojection ofVonto ¥;2", Exercise 2.33*. (a)Interms ofrepresentations VandWinR(G), theinner product onCaq(G) takes thesimple form (VW) =dim Homa(¥, W) 24 2.Characters (b)UeCayu(G)isavirtualcharacter,and(x,2)=1,theneitheryor—1 isthe character ofanirreduciblerepresentation, theplussignoccurringwhen x(1) >0.f(r, x)=2,and x(I) >0,then iseither thesum orthedifference oftwo irreducible characters. (©)IfU, V,and Wareirreducible representations, show that Uappears in V@W ifand only ifWoccurs inV*@U.Deduce that this cannot occur unless dim U>dim Widim ¥. We conclude this lecture with some exercises that usecharacters towork ‘utsome standard facts about representations. Exercise 234", Let Vand Wbeirreducible representations ofG,and Lo:V-+ Wanylinear mapping. Define L:V-+Wby 1 Le)= YoLo(a"o =Ge bolo) Show that L.=OifVandWarenotisomorphic,andthatL.ismultiplication bytrace(LoVdim(¥) ifV=W. Exercise2.35*,Showthatiftheirreducible representations ofGarerepresentedbyunitarymatrices[ofExercise1.14],thematrixentriesoftheserepresenta-tions form anorthogonal basis forthespace ofall functions onG[with inner product given by(2.11)}. Exercise 2.36*. IFG,and G,aregroups, and V,and Varerepresentations of G,andG;,then thetensor product ¥;@V,isarepresentation ofG,xGs, by(a1Xga)-(01 @ea)=940) @ga03.Todistinguishthis“external”tensor product from theinternal tensor product—when G,=G,—this external tensor product issometimes denoted V,@V3.If%isthecharacter ofVj,then thevalue ofthecharacter zofV, V4isgiven bytheproduct: (9s *92)=t1(9s)x2(G2)- ICV, and V,areirreducible, show that VV, isalso irreducible andshow thatevery irreducible representation ofG,xGyarises thisway. Interms of representation rings, RG, xG,)=RG) @RG,). IntheselectureswewilloftenbegivenasubgroupGofagenerallineargroupGL(V),andwewilllookforotherrepresentations insidetensorpowers‘of¥.The following problem, which isatheorem ofBurnside and Molien,showsthatforafinitegroupG,allirreducible representations canbefoundthisway. $24, More Projection Formulas; More Consequences 2 Problem 2.37*. Show that ifVisafaithful representation ofG,ie.p: G-+ GL(¥) isinjective, then anyirreducible representation ofGiscontained in some tensor power V9of¥. Problem 2.38*. Show that thedimension ofanirreducible representation ofGdivides the order ofG. ‘Another challenge: Problem 2.39. Show that thecharacter ofany irreducible representation of dimension greater than 1assumes thevalue 0onsome conjugacy class ofthe group. LECTURE 3 Examples; Induced Representations; Group Algebras; Real Representations ‘This lecture issomethingofagrabbag.Westartin§3.1withexamplesillustratingthe useofthetechniquesoftheprecedinglecture.Section32ialobywayofanexample Wewilseequite abit more about therepresentations ofthesymmetric groups in general later;4sdevotedtothisandwillcertainlysubsumethisdiscussion,butthi ouldprovideaeastasenseofhowwecangoaboutanalyzingrepresentations of ‘clasofgroups,asopposedtoindividualgroups.in3.3and3.4weintroducetwo basinotionsinrepresentation theory,inducedrepresentations andthegroupalgebraFinally in§35 weshow how toclassify representations ofa finite group onareal ‘vector space, given theanswer tothecorresponding question over Cyandsayafew‘wordsabouttheanalogousquestionfrsubfieldsof€otherthanEverythinginthislectureiselementaryexcetExercises39and332,whichinvolvethenotionsofCliflordalgebras and theFourier transform, respectively (both exercises, ofcourse, ean be skipped). S41: Examples: ,and&,§32Exteriorpowersofthestandardrepresentation ofS{23 Induced representations {4 The group algebra $35 Real representations andrepresentations ove subfields of© §3.1. Examples: S;and ‘Wehavefoundtherepresentations ofthesymmetric andalternating groupsforn<4,Beforeturning toamoresystematic studyofsymmetric andalter- nating groups, wewill work outthenext couple ofcases. S21, Examples: Gyand %, n Representations oftheSymmetric Group & Asbefore,westartbylistingtheconjugacy classesof©,andgivingthenumberofelements ofeach:wehave10transpositions, 20three-cycles, 30four-cycles and24five-cycles; inaddition, wehave 15elements conjugate to(12)(34) and10elements conjugate to(12)(345). Asfortheirreducible representations, we have, ofcourse, thetrivial representation U,thealternating representation U', and thestandard representation V;also, asinthecase of,wecan tensor thestandard representation Vwith thealternating onetoobtain another irreducible representation V’with character xy-=xyXu. Exercise 3.1.Find thecharacters oftherepresentations Vand V";deduce in particular that and V”aredistinct irreducible representations. ‘Thefirstfourrowsofthecharacter tablearethus 1 0 m7 0 2 1s 20 Ss | ay ary 039 24) BH (19045) ufi 14 1 1 1 1 a 1 1 “1 vj4ao2 4 0 -t ° = wlao 2 4 a) ° 1 Clearly, weneedthreemoreirreducible representations. Whereshouldwelookforthese?Onthebasisofourpreviousexperience andProblem2.37),natural place would beinthetensor products/powers oftheirreducible representations wehave found sofar,inparticular inV@V(theother twopossibleproducts willyieldnothingnew:wehaveV'@V=V@V@U' andV'@V' =V@V).Ofcourse, V®Vbreaksupinto\?VandSym*¥,sowe look atthese separately. Tostart with, bytheformula Xara) =Marla)? ~x00") ‘wecalculate thecharacter of A?¥: Anew =(6,0,0,0, by~2,0} weseefromthisthatitisindeedafifthirreducible representation (andthatAVQU"=A’,sowegetnothingnewthatway),‘Wecannowfindtheremaining tworepresentations ineitheroftwoways.First,ifn,andn,aretheirdimensions, wehave Sha 120— 14184 EH E, s0nf4+n}=50.Thete arenomore one-dimensional representations, since these aretrivial onnormal subgroups whose quotient group iscyclic, andW, 283.Examples;InducedRepresentations; GroupAlgebras;RealRepresentations istheonlysuchsubgroup. Sotheonlypossibility ism,=ny=5.LetWdenoteconeofthese five-dimensional representations, andsetW”’=W'® U’.Inthe table, iftherowgiving thecharacter ofWis, (oa ay as ty as ae), thatofWis (5a) a,~ay ayay ~ay) Using theorthogonality‘relationsor(2.20),oneseesthatHV’#W;andwithalitlecalculation, uptointerchanging WWandW’,thelasttworowsareasgiven: 1 10 2» 9 24 Is 20 S [toy (yy) 1]BH__ 145) vit 1 0 7 7 1 7 Cr 1 1 “1 vlqoo2 4 0-1 0 - via 4 o = 0 1 a er) ° 1 -2 0 wis obo -t = 0 1 1 wls -t 0-1 1 ° 1 “1 From thedecomposition V®U=C',wehave also*V=ASCS =U', andV*=V.Theperfect pairing’ VxAV4MV =U, taking0x(04©v2A03)10.0AoyAvzAvyshowsthatA\*Visisomorphic toV*@U' =V. ‘Another way tofind therepresentations Wand W’would betoproceed withouroriginal plan, andlookattherepresentation Sym?¥. Wewillleavethisintheformofan exercise: Exercise 32.(i)Find thecharacter oftherepresentation Sym?¥. (ii)Without using anyknowledge ofthecharacter table ofSs,usethisto show thatSym?V isthedirect sumofthree distinct irreducible representations.(ii)Usingourknowledge ofthefirstfiverowsofthecharactertable,showthatSym?Visthedirectsumoftherepresentations U,V,andathirdirreduc-iblerepresentation W.Complete thecharacter table forSs, Exercise 33.Find thedecomposition into irreducibles oftherepresentations AW, Sym, andV@W. 21YardWaren-dimensional vectorspaces,andUisonedimensional aperfectpavingisbilinear mapfsVxH+Uauchtatnononzerovectorvin¥hasfe,)=0.Equivalent, themap¥=Hom(W,U)=1@U,vtfl,wh,ianiomorphism, $3.1. Examples: Syand%, 2» Representations oftheAlternating Group %, What happens totheconjugacy classes above ifwereplace G,by®,? Obviously, alltheodd conjugacy classes disappear; butatthesame time, since conjugation byatransposition isnow anouter, rather than inner, auto- morphism, some conjugacy classes may break intotwo Exercise 3.4Show that theconjugacy class in&,ofpermutations consistingofproductsofdisjointcyclesoflengthsby,ba,...willbreakupintotheunion oftwo conjugacy classes inW,ifallthehyareodd and distinct ifany byare even orrepeated, itremains asingle conjugacy class in%,.(We consider a fixed point asacycle oflength 1.) Inthecase ofMg, this means wehave theconjugacy class ofthree-cycles {asbelore, 20elements), and ofproducts oftwo disjoint transpositions (15 elements); theconjugacy class offive-cycles, however, breaks upinto theconjugacy classesof(12345) and (21345), each having 12elements. {Asforthe representations, theobvious first place tolook isatrestrictionstoML,oftheirreducible representations ofSsfoundabove.Anirreduciblerepresentation of©,may become reducible when restricted toWs; ortwo distinct representations may become isomorphic, aswill bethecase with U and U', Vand ¥',orWand W’. Infact, U,V,and Wstay irreducible since theit characters satisfy (z,x)=1.Butthecharacter ofA?V hasvalues (6,0, =2,1,1)ontheconjugacy classes listed above, so(x,x)=2,and \°V isthesumoftwoirreducible representations, which wedenote byYandZ,Since thesums ofthesquares ofall thedimensionsis0,(dim¥)?+(dimZ)?=18, soeach must bethree dimensional. Exercise 3.5.Usetheorthogonality relations tocomplete thecharacter table ofMy: 120Is 2 2ay|102) 8 (aM) ats4s) uli T 1 1 via o “1 -1 wis 1 1 - %% refs 1-5 y}s0-taon 1-3 148 zis0 1 Live bas ‘Therepresentations ¥andZmayinfactbefamiliar: canberealizedasthegroupofmotionsofanicosahedron (or,equivalently, ofadodecahedron) 303.Examples;InducedRepresentations; GroupAlgebras;RealRepresentations and ¥isthecorresponding representation. Note thatthetworepresentations {W,+GLs(R) corresponding toYand Zhave thesame image, but(asyou ‘eanseefrom thefactthattheir characters differ only ontheconjugacy classes‘of(12345)and(21345))differbyanouterautomorphism ofWs.Notealsothat/\?Vdoesnotdecompose overQ;wecouldseethisdirectlyfrom the fact that the vertices ofadodecahedron cannot allhave rational coordinates, which follows from theanalogous factforaregular pentagon in, theplane. Exercise36,Findthedecomposition ofthepermutation representation of&,‘corresponding tothe(i)verticesi)faces,and(i)edgesoftheicosahedron. Exercise 37.Consider thedihedral group Dz, defined tobethegroup of isometriesofaregularn-gonintheplane.LetI”=Z/n<Dy,bethesubgroup ofrotations. Usethemethods ofLecture |(applied there tothecase Sy=Dg) toanalyze therepresentations ofDy thatis,restrict anarbitrary represemta-tionofDy,toI,breakitupintoeigenspaces fortheactionofF,andaskhowtheremaining generator ofDz,actsofthese eigenspaces Exercise3.8.Analyzetherepresentations ofthe dihedral group D,,using the character theory developed inLecture 2. Exercise 39.(a)Find thecharacter table ofthegroup oforder 8consisting of thequaternions {41, +i,tj,£k} under multiplication, This isthecase m= 3ofacollection ofgroups oforder 2",which wedenote H,Todescribe them, letGydenote thecomplex Cilford algebra generated by0y,..., 0with felations of=—1andv,-0j=—0j-0j,80Cyhasabasisvy=04°...88 1={iy <*"*<i)vaties over subsetsof(1,..",m).(See§20.1fornotationand basic facts about Clifford algebras). Set Hy,={bop [Miseven) <(gy This group isa2-to-1 covering oftheabelian 2-group ofmxmdiagonal matrices with 1diagonal entries and determinant 1.The center ofHyis {£1} ifmisoddandis{1, £0)),__)} if'miseven. Theother conjugacy classes consist ofpairs ofelements {0,). The isomorphismsofCz*"witha matrixalgebra oraproduct oftwo matrix algebras give a2-<dimensional “spin” representation SofHq, andtwo2"-dimensional “spin” or“half- spin” representationsS*and5”ofHy (b)Compute thecharacters ofthese spin representations andverify that they areirreducible, (©)Deduce that thespin representations, together with the2"-' one- dimensional representations coming from theabelian group H_/{ +1)givea ‘completesetofirreduciblerepresentations, andcomputethecharactertable forHy $32. Exterior Powers ofthe Standard Representation ofS, a For oddmthegroups11,areexamplesofextra-special 2-groups,cf.(Grie], [Qu]. Exercise 3.10, Find thecharacter table ofthe group SL3(Z/3), Exercise 3.11. Let11(2/3) betheHeisenberg group oforder 27: lab HEB)=4{0 1¢|abeez/3} <SLs(2/3). oot ‘Analyze therepresentations ofH(Z/3), first bythemethods ofLecture 1(restricting inthiscasetothecenter 106 z=4{0 1o|vezap=zs oot ofH(Z/3)), and then bycharacter theory. §3.2. Exterior Powers oftheStandard Representation ofS How should wegoabout constructing representations ofthesymmetric groups ingeneral?Theanswertothisisnotimmediate;itisasubjectthatwill occupy most ofthenext lecture (where wewill produce alltheirreduciblerepresentations of©,).Fornow,asanexampleofthe elementary techniquesdeveloped sofarwewillanalyzedirectlyoneoftheobviouscandidates: Proposition 3.12.EachexteriorpowerM*Vofthestandardrepresentation Vof&,isirreducible, 0.<k sd ~1. Proor. From thedecomposition C4=V@U,weseethatVisirreducible ifandonlyif(es,Zea)=2.Similarly,since NC =(AV @NU) O(N V@AU) =NVONY, itsufficestoshowthat(x,x)=2,wherezisthecharacteroftherepresentationNCELetA={1,2,...5d). ForasubsetBofAwithkelements,andg¢G= Sy let 0ifgm 4B {ole=4 1ifg(B)=Bandglyisanevenpermutation =1 ifo(B)=Bandglyisod. 323, Examples; induced Representations; Group Algebras Real Representations Here, ifg(B) =B,glydenotes thepermutation ofthesetBdetermined byg. ‘Then x(4)=E{a}s, and 1 2 w=7(Fon) 1 =ALEPWala 1=GYFDenale)(senglo, where thesums areover subsets Band CofAwith kelements, and inthelast equation, thesum isover thosegwthg(B)=Bandg(C)=C.Suchgisgiven byfourpermutations: oneofByC,oneofB\B>C,oneofC\B-C,andone ofA\BuC,Letting Ibethecardinality ofB>C,thislastsumcanbewritten 1 i(sgna)?(sgnb)(sgnc) TEE FiskscehsteEig OPONERD I =tyyma- 2 «).appre a0006.%so)(Xv) Theselastsumsarezerounlessk—1=0or1.Thecasek=|gives L 1(a ty a—met —wl=ayh@-b a(sjelat. Similarly, theterms withk—|=|alsoaddupto1,so(x,x)=2,asrequired. a Notebywayofcontrast that thesymmetric powers ofthe standard repre- sentation ofS,arealmost never irreducible. Forexample, wealready know thattherepresentation Sym?Vcontains onecopyofthetrivialrepresentation:thisisjustthestatement thatevery irreducible realrepresentation (such asV) ‘admits aninner product (unique, uptoscalars) invariant under thegroupaction;noristhequotient ofSym?Vbythistrivialsubrepresentation neces- sarily irreducible, aswitness thecase ofGs, §3.3. Induced Representations ICH©Gisasubgroup,anyrepresentation VofGrestrictstoarepresentationofH,denotedResfVorsimpleResV.Inthissection,wedescribeanimpor-tantconstruction whichproduces representations ofGfromrepresentationsofH.Suppose Visarepresentation ofG,and W<Visasubspace which isH-invariant. ForanyginG,thesubspace gW={g:w:weW}dependsonlyontheleftcosetgHofgmoduloH,sincegh:W=g-(h-W)=g:W;foracoset $33. Induced Representations 3 in G/H, wewrite0WforthissubspaceofV.WesaythatVisinducedbyW ifevery element inVcan bewritten uniquely asasum ofelements insuch translates ofW,ic, ve Ww InthiscasewewriteV=IndgW=IndW. Example 3.13. Thepermutation representation associated tothe leftaction of GonG/H isinduced from thetrivial one-dimensional representation WofH.HereVhasbasis{e,:0€G/H},andW=C-ey,withHHthetrivialcoset. ‘Example 3.14. The regular representation ofGisinduced from theregular representation ofH.Here Vhasbasis {e,:g€G}, whereas Whasbasis {eyheH}. ‘Weclaim that, given arepresentation Wof11,suchVexistsandisunique uptoisomorphism, Although wewilllater give several fancier ways tosee this, itisnothard todoitbyhand. Choose arepresentative g,€Gforeach coset 0€G/I1, with erepresenting thetrivial coset H¥.Toseetheuniqueness, note that each element ofVhasauniqueexpression v=Y°g,,,forelementsw,inW.GivenginG,writegg,=g,-hforsome €G/ilandheHl.Then ‘we must have 9°(9.%) =(9°90)%s=(OeWms=ah). This proves theuniqueness and tells ushow toconstruct V=Ind(W) from W.Take acopy W’of foreach leftcoset o€G/H1; forweW,letg,wdenote theelement ofW"corresponding towinW.Let¥=GP)W*,soevery element ofVhasauniqueexpression v=J9,w,forelementsw,inW.Given9€G, define (92%) =9.0) if9-9, =ah. ‘Toshow that this definesasactionofGonV,wemustverifythatg’-(q-(geW.))=(0'-4)-(ge,) foranother element g'inG.Now ifg'-g, =9,°H, then 0'-(0-(G4™,))=9°(axhw,))=94(Hthw,)). Since(9°-9):4.=9"°(g"_)=9"G."h=gp"W-h,wehave (6'-0)-(900,) =9gl(HA)m,)=a,(H!-Ch,)) asrequired. Example 3.5. IfW=(W;, thenIndW=(Ind W,. ‘The existence oftheinduced representation follows from Examples 3.14 and3.15since any Wisadirect sum ofsummands ofthe regular representation. 343.Examples;InducedRepresentations; GroupAlgebras;RealRepresentations Exercise 3.16. (a)IfUisarepresentation ofGand Warepresentation ofH, show that (with allensor products over ©) U@ Ind W=Ind(Res(U) @W), Inparticular, Ind(Res(U)) =U@P,where Pisthepermutation representationofGonG/H.ForaformulaforRes(Ind(W)), forWarepresentation of1,see[Se2, p.8] (b)Like restriction, induction istransitive: ifHc K&Garesubgroups, show that Ind§(W) =Ind{(Ind§ W). Note that Example 3.15 says that themap Indgives agroup homomor-phismbetweentherepresentation ringsR(H)andR(G),intheoppositediree-tion from thering homomorphism Res: R(G) -+R(H) given byrestriction;Exercise3.16(a)saysthatthismapsatisfiesa“push-pull” formula2:Ind(f)= Ind(Res(a)-) with respect totherestriction map. Proposition 3.17. LetWbearepresentation ofH,Uarepresentation ofG,and suppose V=IndW.Then any H-module homomorphism g:W-+ Uextends uniquely t0aG-module homomorphism @:V-+ U.ie, Hom(W, ResU)=Hom,(Ind W,U) Inparticular, this universal property determines IndWuptocanonical ‘somorphism. Proor.WithV=@,ano:Wasbefore,define@ono°Wby a:Wt. Ws Uo U, which isindependent oftherepresentative g,forosince @isH-linear. Tocompute thecharacter ofV=IndW,notethatg€GmapsoWtogoW, 50thetrace iscalculated from those cosets @with go=0,ie,sgse HTfor sea. Therefore, Kiva) =YEw(s"'9s) (8€oarbitrary). (3.18) Exercise 3.19. (a)IfCisaconjugacy class ofG,and CH decomposes into conjugacy classes D,,..., D,ofH,(3.18) can berewritten as:thevalue ofthe character ofInd WonCis IGl <.1D inar()=FaFetol) Yoel=iBjc (6)IfWisthetrivial representation ofH,then (GH),naw) =oD canraat=Fe, ICOHL §33. Induced Representations 8 Corollary3.20(Frobenius Reciprocity). IfWisarepresentation ofHandUarepresentation ofG,then (Chaney Lode =(ors Ineswdn Proor. Itsuffices bylinearity toprove thiswhen Wand Uareirreducible. The left-hand side isthenumber oftimes Uappears inIndW,which is thedimension ofHome(Ind W,U).The right-hand side isthedimension of Homy(W, ResU).These dimensions areequal bytheproposition o IWandUareirreducible,Frobeniusreciprocitysays:thenunheroftimes Uappears inIndWisthesame asthenumber oftimes Wappears inResU.Frobenius reciprocity canbeusedtofindcharacters ofGifcharacters ofHare known. Example 321, Wecompute IndjW, when H=,<G=Sy,W=Va(the standard representation) =U3(thealternating representation). Weknow the irreducible represenatations ofS4:Us,Us,Vs,which restrict Us,Us=Vay U;®Us,respectively. Thus, byFrobenius, Ind Vy=U;®Vs, Example 222. Consider next H=S¢G=G,, W=Vs.Again weknow the irreducible representations,andResUs=Us,ResU;=Uj,ResVy=Uy®Vs [the vector (Le By=3)©Va={p25X354)!D1=OFisfixedby11],ResVj=Uy®V5,withVj=V,,andResW,=V,(asonemayseedirectly), Hence, IndVs=Ve@ ¥@W,.(Note that theisomorphism ResW,=¥,actually follows, since one W,isallthat could beadded to Ve@ Veto getInd V4.) Exercise 323, Determine theisomorphisin classes ofthe representationsofS, induced by()theone-dimensional representation ofthegroup generated by (1234) inwhich (1234)-v =io,i=/—T;(i)theone-dimensional representa tionofthegroup generated by(123) inwhich (123)-0 =e", Exercise324.LetH=Wy¢G=&,.ShowthatIndU=U@U',Ind¥=V@ V',andInd W=W@ W’,whereas Ind¥=IndZ=AV. Exercise 3.25*. Which irreducible representations of©,remain irreducible when restricted to%,? Which areinduced from 91,? How much does thistell ‘youabout theirreducible representations of2,2 Exercise 3.26*. There isaunique nonabelian group oforder 21,which canberealizedasthegroupofafinetransformations x1-»ax+Bofthelineoverthe fieldwithsevenelements,withaacuberootofunityinthatfield,Findtheirreducible representations andcharacter table forthisgroup. 36 3.Examples; Induced Representations; Group Algebras; Real Representations Nowthatwehaveintroduced thenotionofinducedrepresentation, wecanstate two important theorems describing thecharacters ofrepresentations ofAfinitegroup.Intheprecedinglecturewementioned thenotionofvirtualcharacter;thsisjustanelementofthe image Aofthecharacter map ERG) +ConulG) from therepresentation ring R(G) ofvirtual representations. The following twotheorems both state thatinorder togenerate A@)Q(resp. A)itis enoughtoconsider thesimplestkindofinducedrepresentations, namely,thoseinducedfrom eyclic (respective elementary) subgroups ofG.Fortheproofs ofthese theorems werefer to[Se2, §9,10]. Wewillnotneed them inthese lectures. Attin’s Theorem 3.27. Thecharacters ofinduced representations from eyclie subgroups ofGgenerate alattice offinite index inA. Asubgroup HofGisp-elementary ifH=AxB,with Acyclic oforder prime topandBap-group. Brauer’sTheorem3.28.Thecharactersofinducedrepresentations fromelemen-tary subgroups ofGgenerate thelattice A. §3.4. The Group Algebra There isanimportant notion that wehave already dealt with implicitly butnotexplicitlythisisthegroupalgebraCGassociated toafinitegroupG.Thisisanobject thatforall intents andpurposescancompletelyreplacethegroup Gitself;anystatementabouttherepresentations ofGhasanexactequivalentstatement about thegroup algebra. Indeed, toalarge extent thechoice oflanguageisamatteroftaste.‘Theunderlying vectorspaceofthegroupalgebraofGisthevectorspacewith basis (e,}corresponding toelements ofthegroup G,thatis,theunder- lying vector space oftheregular representation. Wedefine thealgebra struc- tureonthisvector space simply by et ee Byarepresentation ofthealgebra CGonavector space Vwemean simply analgebra homomorphism CG+End(v), 50thatarepresentation VofCGisthesamethingasaleftCG-module, Note thata representationp:G-+Aut(V)willextendbylinearitytoamapj:CG-» End(V),s0 thatrepresentations ofCGcorrespond exactly torepresentations ofG;theletCG-module given byCGitself corresponds totheregular representation. 24. TheGroup Alben ” If(19)aretheirreducible representations ofG,thenwehaveseenthattheregular representation Rdecomposes R=Qemjownm Wecannow refine thisstatement interms ofthegroup algebra: wehave Proposition 3.29. Asalgebras, 66=@End) root.Aswehavesad,foranyrepresentation WofG,themapG->Aut(H¥)extends bylinearity toamap CG~»End(W); applying thistoeach ofthe irreducible representations W,gives usacanonical map 9:06-+@ Endy ‘This isinjective since therepresentation onthe regular representation isfaithfulSincebothhavedimension5(im1),themapisantomorphism, 0 Afewremarks areinorder about theisoinorphism oftheproposition. First, 9canbeinterpreted astheFourier transform, ef.Exercise 3.32. Note also that Proposition 2.28 hasanatural interpretation interms ofthegroupalgebra:itsaysthatthecenterofCGconsistsofthoseJ"a(g)e,forwhichais class function. Next,wecanthinkofgasthedecomposition ofthe semisimple algebra CG into aproduct ofmatrix algebras. 1implies thatthe matrix ences ofthe irveductble representations give abass forthe space ofall functions onG,cf. Exercte 235, 'Note inparticular that any reducible representation isomorphic toa (evnimal) letideal inCG. These letideals aregenerated byidempotents. hn fact, wecan interpret theprojection forma ofthe astlecture inthelanguage ofthe group algebra: theformulas sysimply that theelements — dim w-L iwlah-e, 806Gide aretheidempotents inthegroup algebra corresponding tothedirect sum factors inthedecomposition ofProposition 3.29. Tolocate theirreduciblerepresentations W,ofgroupG[notjustadirectsumofdim(1¥,)copies),we‘want tofind other idempotents ofCG. Wewillseethiscarried outfortheSymmetricgroupsinthefollowinglecture“Thegroupalgebraalsogivesusanotherdescription ofinducedrepresentations ifWisarepresentation ofasubgroup 11ofG,then theinduced representation may beconstructed simply by Ind W= C6Ben 38 3.Examples: Induced Representations; Group Algebra; Real Representations ‘whereGactsonthefirstfactor:g(ey®w)=é[email protected] of thereciprocity theorem isthen aspecial case ofageneral formula forachange ofrings CH ~+C6: Homen(W, U)=Homeg(CG @cu W,U). Exercise 3.30". Theinduced representation Ind(W) canalso berealized con- cretely asaspace ofW-valued functions onG,which canbeuseful toproduce matrix realizations, orwhen trying todecompose Ind(W) into irreducible pieces. Show that Ind(W) isisomorphic to Homp,(CG, W)={f:G+W:f(hg) =hf(a),VheH,geG}, where Gacts by(@'-f(a) =S(aa’) Exercise 3.31.IfCGisidentified withthespaceoffunctions onG,thefunction‘corresponding toS'sq@(g)é,, show thattheproduct inCGcorrespondstotheconvolution +offunctions: (9+Wa)=5othwiirtoy. (With integration replacing summation, thisindicates how onemay extend thenotion ofregular representation tocompact groups.) Exercise 332°. Ifp:G+GL(V,)is arepresentation, and9isafunctiononG, definetheFourier transform @(p)inEnd(¥,) bytheformula 10)=F000). {@)Show that9°¥(0)=@l0)910). (b)Prove theFourier inversion formula 1 10)=[GXdiemVe)Trace((a")- (0), thesumovertheirreducible representations pofG.Thisformulaisequivalent toformulas (2.19) and(2.20). (©)Prove thePlancherel formula forfunctions »and yonG: oy 1 ¥,eCWa)=Gydion) TracetHpH(0) Ourchoiceofteft action ofagroup onaspace hasbeen perfectly arbitrary,andtheentirestoryisthesameiGaetsontherightinstead,Moreover,thereisastandard way tochange aright action into aleftaction, and viceversa: Given aright action ofGon¥,define theleftaction by goemv(g') geGveVv. $35. Real Representations and Representations over Subfieds ofC 2» IfA=CGisthegroup algebra, aright action ofGonVmakes Varight A-module, Toturn right modules into leftmodules, wecan usetheanti- involution a+-+4 ofAdefined by(5.4,¢,)" =Saye, ArightA-module is then turned intoaleftA-module bysettinga0=v= Thefollowingexercisewilltakeyoubacktotheoriginsofrepresentationtheory inthe19th century, when Frobenius found thecharacters byfactoring, this determinant. Exercise3.33*.GivenafinitegroupGofordern,takeavariablex,foreachelement ginG,and order theelements ofGarbitrarily. LetFbethedeter-rminantofthe nxnmatrix whose entry intherow labeled bygand columnlabeledbyhisx,4-1.Thisisaformofdegreeminthenvariablesx,,whichisindependent oftheordering. Normalize thefactors ofFtotake thevalue 1 when x,=1and x,=0forg#e. Show that theirreducible factors ofFcorrespond totheirreducible representations ofG.Moreover ifFisthefactor correspondingtotherepresentation p,showthatthedegreeofF,isthedegree 4(p)oftherepresentation p,andthateach F,occurs inFdp)‘times. fx,is thecharacter ofp,show thatz,(g) isthecoefficient ofx,x4" inF,, §3.5. Real Representations and Representations over Subfields ofC Ifagroup Gacts onareal vector space Vo,then wesaythecorresponding, ‘complex representation ofV=Vo®q€isreal.Totheextentthatweare interested intheaction ofagroup Gonreal rather than complex vectorspaces,theproblemwefaceistosaywhichofthecomplexrepresentations ofGwehave studied are infact real Our first guess might bethat arepresentation isreal ifand only ifitscharacteris real-valued. This(urnsoutnottobethecase:thecharacter ofatealrepresentation iscertainlyreal-valued, buttheconverseneednotbetrue.Tofindanexample,supposeG<SU(2)isafinite,nonabelian subgroup.ThenGactsonC?=Vwith areal-valued character since thetrace ofanymatrixinSU(Q)isreal.IfVwerearealrepresentation, however,thenGwouldbeasubgroup ofSO(2) =$",which isabelian. Toproduce such agroup, notethat SU(2)canbeidentifiedwiththeunitquaternions. SetG=(1,+i,+i,+h). Then G/{+1}isabelian, sohasfourone-dimensional representations, which sive four one-dimensional representations ofG.Thus, Ghasoneirreducible {wo-dimensional representation, whose character isreal, butwhich isnot real Exercise 334%, Compute thecharacter table forthisquaternion group G,and compare itwith thecharacter table ofthedihedral group oforder 8 403, Examples; Induced Representations; GroupAlgebras;RealRepresentations ‘Amore successful approach istonote that ifVisarealrepresentation of G,coming from Voasabove, then one canfind apositive definite symmetric bilinear forin on¥,which ispreserved byG.This givesasymmetricbilinear form onVwhich ispreserved byG.Not every representation will have such aform since degeneracies may arise when one tries toconstruct one following theconstruction ofProposition 1.5.Infact, Lemma 3.35.Anirreducible representation VofGisrealifandonlyifthereis ‘anondegenerate symmetric bilinear form BonVpreserved byG. Proor. Ifwehave such B,andanarbitrary nondegenerate Hermitian form H, also G-invariant, then vavedy gives aconjugate linear isomorphism @from ¥toV:given xeV,there isa unique (x)€FwithB(x,y)=H(p(x),y),and@commuteswiththeaction ofG.Then 9?=p @isacomplex linear G-module homomorphism, so g?=Ald. Moreover, (OC), »)=Bix,¥)=BLY.) =H((). 9)=HG, 00). from which itfollows that H(g?(x), y)=H(x, @(y)), andtherefore 2isa positive real number. Changing Hbyascalar, wemay assume =I,so‘p?=Id.Thus,¥isasumofreal eigenspaces V,and V_for@corresponding. toeigenvalues |and —1.Since »commutes with G,V,and V_areG-invariant subspaces. Finally, p(ix) =—ig(x), soiV,=V.,and V=V,@C. a Notefromtheproofthatarealrepresentation isalsocharacterized bythe‘existence ofaconjugate linear endomorphism ofVwhose square isthe identity; ifV=Vo@qC,itisgiven byconjugation: v9@A+v9@A. Awarning isinorder here: anirreducible representation ofGonavector ‘space over Rmay become reducible when weextend thegroup field toC.To ivethesimplest example, therepresentation ofZ/nonR?given by Onk 2nk' cos sin tay° 2nk 2nk sins osm isirreducible over Rforn>2(nolineinR?isfixed bytheaction ofZ/n), but willbereducible over C.Thus, classifying theirreducible representations ofG coverCthatarerealdoesnotmeanthatwehaveclassifiedalltheirreducible realrepresentations. However, wewillseeinExercise 3.39 below how tofinish thestory once wehave found therealrepresentations ofGthatareirreducible over C. 415, Real Representations andRepresentations over Soild of© “ SupposeVisanirreduciblerepresentation ofGwithz,real.Thenthereis 4G-equivariant isomorphism Vx V*,ie,there isaG-equivariant (non- degenerate) bilinear form BonV;but,ingeneral, Bneed notbesymmetric. Regarding Bin V*@V* =Symv* @AVS, andnoting theuniqueness ofBuptoinultiplication byscalars, weseethat B iseithersymmetric orskew-symmetric, IfBisskew-symmetric, proceeding as above one canscale sog?=—Id. This makes V“quaternionic,” with @ becoming multiplication? byj: Definition 3.36.Aquaternionic representation isa(complex) representation V which hasaG-invariant homomorphism J:V+ Vthat isconjugate linear, ‘andsatisfies J?=—Id. Thus, askew-symmetric nondegenerate G-invariantBdetermines aquaternionic structureon¥. Summarizing thepreceding discussion wehave the ‘Theoem 337. Anirreducible representation VIsone and only one ofthe {following (1)Complex: zyisnotreal-valued; Vdoes nothave aG-invariant non- degenerate bilinear form. (2)Real: V=Vo®C, areal representation; VhasaG-invariant symmetric nondegenerate bilinear form. (3)Quaternionie: zyisreal, butVisnotreal; VhasaG-invariant skew- symmetric nondegeneate bilinear form. Exercise 3.38. Show that forVirreducible, 0ifVis complex 1ma (g?)=1ifVisreal wae —1 ifVisquaternionic. “This verifies thatthe three cases inthetheorem aremutually exclusive. Italso implies that iftheorder ofGisodd, allnontrivial representations must be ‘complex, Bxercise 3.39. Let Vpbeareal vector space onwhich Gacts irreducibly,V=¥o@Cthecorresponding realrepresentation ofG.ShowthatifVisnotirreducible, then ithasexactly twoirreducible factors, andthey areconjugate ‘complex representations ofG. +See12formor onqunernion andquteroni epresenaton, 423. Examples; Induced Representations; GroupAlgebras;RealRepresentations Exercise3.40,Classifytherealrepresentations of2. Exercise3.41*.ThegroupalgebraRGisaproductofsimpleR-algebras corre-spondingtotheirreducible representations overR.Thesesimplealgebrasaretmatrix algebras over C,R,orthequaternions Haccordingastherepresenta tion iscompler, real, orquaternionic. Exercise 342°. (a)Show that allcharacters ofagrouparerealifandonlyif every element isconjugate toitsinverse.(b)Showthatanelement¢inasplitconjugacyclassof®t,isconjugateto itsinverseifandonlyifthenumberofcyclesin¢whoselengthiscongruentt03 modulo 4iseven. (©)Show thattheonly d’sforwhich every character ofis real-valued are d= 1,2,5,6,10, and 14, Exercise 343°. Show that: ()thetensor product oftworealortwoquater- niionic representations isreal; (i)forany ¥,V*®Visteak (i)ifVis real, so areall/'V; (iv)ifVisquaternionic, AYisrealforkeven, quaternionic for kod. Representations over Subfields ofCinGeneral Weconsidernextthegeneralization oftheprecedingproblemtomoregeneralsubfields ofC.Unfortunately, our results willnotbenearly asstrong in general, butwecan atleast express theproblem neatly interms ofthe representation ring ofG. Tobegin with, our terminology inthis general setting isa little different. LetK<Cbeanysubfield. Wedefine aK-representation ofGtobea vector space Voover Konwhich Gacts; inthiscase wesaythat thecomplex representation V=Vo®Cisdefined over K. One way tomeasure how many oftherepresentations of Garedefined overafieldKistointroduce therepresentation ringRg(G)ofGoverK.Thisis‘defined justliketheordinary representation ring; thatisitisjustthegroupofformallinearcombinations ofK-tepresentations ofGmodulorelationsof theform ¥+W~(V@ W),with multiplication given bytensor product. Exercise344°,Describetherepresentation ringofGoverRforsomeofthegroups Gwhose complex representation wehave analyzed above, Inpartic- ular, istherank ofRq(G) always thesame astherank ofR(G)? Exercise2.45*.(a)ShowthatRe(G)isthesubringoftheringofclass functions conGgenerated(asanadditivegroup)bycharacters ofrepresentations definedover K. §35. Real Representations andRepresentations over Subic ofC “a (b)Show that thecharacters ofirreducible representations over Kform an ‘orthogonal basis forRx(G)-(€)Showthatacomplexrepresentation ofGcanbedefinedoverKifandonly ifits character belongs t0Ry(G)- Formore ontherelation between Ry(G) andR(G), see[Se2]. LECTURE 4 Representations ofS,:Young Diagrams and Frobenius’s Character Formula Jnthislecture weget10work. Specifically, wegive in§4.1acomplete description oftheieredociblerepresentations ofthesymmetricgroup,thatis,aconstuction oftherepresentations (viaYoung smmetrizers) andaformaFrobenive’formula)forheir characters. The proof thatthe representations constructed in41 are indeed the inreducible representations ofthesymmetric group isgiven in642; theproof of Frobenius formula, aswellasanumberof others, in4.3. Apa from thei intiasic interest (and undeniable beauty), these results ura Outtobe ofsubstantial interest in Lietheory analogsoftheYoungsymmetizerswilgiveaconstrctionoftheieredc- iblerepresentations ofSLC. ACthesume time, while thetechniquesofthislectureare completely elementary (weuseonly fewidentities about symmetic polynomial,provedinAppendixA),thelevelofdifclty isclearly higher than inprecedingfecture.Theresisinthelaterhalfof43(romCorollary439on)inparticularae «quite dtc, and inasmuch asthey atenotused later inthe text may beskipped by teaders who arenotsymietic grovp enthusiasts. HA:StatementoftheresultsH2: Ireducible representations ofSy $43 Proof ofFrobenius’ formula §4.1. Statements oftheResults ‘The number ofirreducible representaton ofS;isthenumber ofconjugacyclasses,whichisthenumberp(d)ofpartitions! ofd:d=A,Fo"Ay Ay>BAe Wehave "1c tometns convenient,andsometinesanuisance,fohavepartonsthatendinoneof toreserianveiens,weallsomeoftheontheendtobemr.Twquencesdefine thesame parton,ofcousheydironybyzerosateend 4, Statements oftheRests 43 Enae=f (1, roe Nee ee ee eo | which converges exactly in{t]<I.This partition number isaninteresting arithmetic function, whose congruences andgrowth behavior asafunction of dthave beenmuch stoied (ef.Har}, (And). Forexample, pd)iasymptoi- callyequal to(ad)e"V4, witha=4/3andf=x/3). Toapartition A=(A;,..., 4,)isassociated aYoung diagram (sometimes called aYoung frame orFerrers diagram) ay aa As a As with 2,boxes intheithrow, therows ofboxes ined uponthefet. The conjugate partition X'=(2,,...,4,) tothe partition Aisdefined byinter- changing rows andcolumns intheYoung diagram, ic,reflecting thediagram inthe45°line. Forexample, thediagram above isthat ofthepartitionG,3,2,1,I,whoseconjugate is(5,3,2).(Withoutreferencetothediagram,the‘conjugate partition toAcanbedefined bysaying 2;isthenumber ofterms in thepartition 2that aregreater than orequal toi) Young diagrams canbeused todescribe projection operators forthe regular representation, which willthen give theireducible representations of G,.For agiven Young diagram, number theboxes, sayconsecutively as shown 1[2[3] 4b ES} Moregenerally, defineatableau onagivenYoungdiagram tobeanumberingoftheboxesbytheintegers I,...,d.Given atableau, saythecanonical one shown, define twosubgroups? ofthesymmetric group >ratbeaothr thanthecanoialoeweechen,onewletifentousinactPant Qa dierent ements nthe group Ping teepeenttons contacted way titteSomatic 46 4,Representations ofG,:YoungDiagramsandFrobenius’ CharacterFormula P=P,= (geSgpreserveseachrow} and 0=0,=(9€S,:gpreserveseachcolumn}. Inthegroup algebra CG,, weintroduce twoelements correspondingtothese subgroups:we set a=ZoeandbiFsentaleg ay Toseewhat a,andb,do,observe thatifVisanyvector space and&,acts fonthedthtensor power ®bypermuting factors theimage oftheelement 4,€CG, ~+End(V®)isjustthesubspace Im(a,) =Sym*'V @Sym4V @--@Sym*v cVM, where theinclusion ontherightisobtained bygrouping thefactors ofV°* according totherows oftheYoung tableaux. Similarly, theimage ofb,on thistensor power is Im(,) = MV@NV ONY VE, ‘where jitheconjugate partition to2 Finally, weset gaa helS (42) thisiscalled aYoung symmetrizer. Forexample, when 1=(d),ey=uy= Syeasén andtheimage ofcyonV%isSym‘¥. When A=(I,.-51sGaray=By,...1)=Dene2,88M(O)eq,andtheimageofc,omVisMY.Wewilleventualy seethattheimage ofthesymmetrizers ¢,inV®provide essentially allthefinite-dimensional irreducible representations ofGL(V). Here westate thecorresponding factforrepresentations ofS,: ‘Theorem 4.3.Some scalar multiple ofcyisidempotent, Le,¢2=nycy, andthe image of¢(byright multiplication onC&,) isanirreducible representation V;ofG,.Every irreducible representation ofG,can beobtained inthis way forauniquepartition. Wewill prove this theorem inthenext section. Note that, asacorollary, cach irteducible representationofG,canbedefinedovertherationalnumbers since¢,isintherationalgroupalgebraQG,.Notealsothatthetheoremgives adirect correspondence between conjugacy classes in©,and irreducible representations ofG,,something which hasnever been achieved forgeneral groups. Forexample, for4=(d), HaC8FgeOF {HLL Statements ofthe Results ” isthetrivial representation U,andwhen A=(1,..., 1, Kisco CBsY.sentaley =CF sentode, isthealternating representation U’.For4=(2,1), Cray =(61+eaad“(Er easy)=E+ea2y~easy=Cas2»inCEs,andVa,y)isspannedbyc.2,1)and(13)-¢,1$0Mea,isthestandard representation of5. Exercise 44*. SetA=Cy, 80Vy=Acy=Addy. (a)Show that P= Ab, {b)Show that ¥,istheimage ofthemap from Aa,toAb,given byright‘multiplication byb,.By(a),thisisisomorphic totheimageofAb, +Aagiven byright multiplication bya,. {(c)Using (a)and thedescription ofV,inthetheorem show that Vy=VW@U, where istheconjugate partitiontoAandU"isthealternating representation, Examples 45,Inearlier lectures wedescribed theireducible representations of6,ord <5,From theconstruction ofthe representation correspondingto 4Young diagram itisnothard towork outwhich representations come from which diagrams: & tvs A]atemating 8, oOsUUteva AU"ahemating FP ¥standara S, cmvu|uv a 484 Representations ofG,:YoungDiagramsandFrobenius's CharacterFormula 8s comvufluv ppoyPPvPenv fw apw Exercise 4.6*. Show that forgeneral d,thestandard representation Vcorre- sponds tothepartition d=(d~1)+1,Asachallenge, youcantrytoprove that theexterior powers ofthestandard representation Varerepresented by a“hook”: £ffnv \ NotethatthisrecoversourtheoremthattheA'Vareirreducible Next weturn toFrobenius’s formula forthecharacter z,ofVi,which includes aformula foritsdimension. LetC,denote theconjugacy class inS, determined byasequence TaClgsiasocesig) withYad,=dsGconsistsofthosepermutations thathavei{-cycles,iy2-cyces,...,andiya-eyctes Introduce independent variables x,,..., %with katleast aslarge asthenumberofrowsintheYoungdiagramof4.DefinethepowersumsPix),<j <d,and thediscriminant A(x) by Pye xfead te exh,Feed ek ! an Aly)=q(%—x). If(3)=f(y.---1%4)isFormalpowerseries,and(ly...)isaktuple ofnon-negative integers, let CSOMay. csty=COMMficient ofx]!----- xfinf. (48) GivenapartitionA:Ay2Ay2Ootd,set Wea tko bed tkQh dy 49) $4.1. Statements oftheResults 0 astrictly decreasing sequence ofknon-negative integers.ThecharacterofV; evaluated ong€C;is given bytheremarkable Frobenius Formula 4.10 2G)=[aqT)reerFP Says Forexample, ifd=5,A=(3,2),andC;istheconjugacy classof(12)(345), eyiy=0,iy=1,ig=1,then Has) =[le—23)03 +DOP +haa =b Other entries inourcharacter tables forG5,S,,and Gycanbeverified as easily, verifying theassertions ofExamples 4.5.Intermsofcertainsymmetric functions, calledSchurpolynomials, Fro-benius's formula canbeexpressed by T]Bea=Ens, thesumoverallpartitions 2ofdinatmostkparts(ef.Proposition 437and (A.27)). Although wedonot use Schur polynomials explicitly inthis lecture, they piay thecentral role inthealgebraic background developed in Appendix A. LetususetheFrobenius formula tocompute thedimension of¥;.The conjugacy class oftheidentity corresponds toi=(d),so im Vy=alGig) =(BC)52Aa Now A(s) istheVandermonde determinant: Toyo apt ig ©=F,amo)tomags ‘The other term is a ererte thesum over k-tuples (r,,-..,1%) that sum tod.Tofind thecoefficient of xi'-,,.-xi¢ intheproduct,wepairoffcorresponding termsinthesetwosums, geting a Leenaay eho OT thesum over those oinG,such that sy —o(f) +£2Oforall t<i<k ‘This sum can bewritten as 504 Representations ofYoungDiagramsandFrobeniusCharacterFormula a .iiniid,8LNUh=Yr. ~ok~f+ 42) a'4MM » ieAer] .a Tee Bycolumn reduction thisdeterminant reduces tothevanderMonde deter- ininent 30 dt Kye Tat, 4.10 dimY=FigLJ9 ay with b=yeh ‘There isanother way ofexpressing thedimensions oftheV;.The hook length ofaboxinaYoungdiagram isthenumber ofsquares directly below ordirectly totheright ofthebox, including theboxonce. Inthefollowing diagram, each boxislabeled byitshook length: {efaq3qi) [aj2hi] OJ Hook Length Formula 4.12. a dimv,=im"4=TTHooklengths) Fortheabovepartition 4+3+1of8,thedimension ofthecorresponding representation ofG,istherefore 81/6-4:4:2-3 =70. Exercise 4.13*.Deducethehooklengthformula fromtheFrobenius formula aan Exercise 4.14*,Usethehooklengthformula toshowthattheonlyirreducible representations ofS,ofdimension lessthan darethetrivial and alternating representations Uand U"ofdimension I,thestandard represetation F and V’=V@U' ofdimension d—1,and three other examples: the(wo- dimensional representation of©,corresponding tothepartition 4=242, and thetwo five-dimensional representations ofSgcorresponding (othe pautiions@= 3h3and624242 41,StatementsoftheResults Fa Exercise4.15*.UsingFrobenius's formulaorotherwise, showthat Ha-1.Cd =ty—Ny Haa-2.1.(G) =Mi=Mh=2)=tas Na-aalG) =Mi—WG=2)+=1 Can you continue this list? Exercise 4.16*. IfgisacycleoflengthdinG,,showthaty,(g)is+1ifLisa hook, and zero ifAisnot ahook: HIdG. DOsssd-t x0ETeae ST Exercise 4.17. Frobenius [Frot] used hisformula tocompute thevalue ofx, ‘onacycleoflengthm<d.{@)Following theprocedure that ledto(411)—which was thecase m= 1—show that dimVz&Wtly) 1a(12.-))=TpYeay (4.18) whereh,,=d!/(d—m)imisthenumber ofcyclesoflengthm(ifm>1),and . = oe)=J]=IWed)=ote—myTPe+1). ‘The sum in(4.18) can berealized asthecoefficient ofx~'intheLaurent expansion ofp(x)/¢p(x) atx=oo. ‘Define therankrofapartition tobethelengthofthediagonal ofitsYoung. diagram, and leta,andb,bethenumber ofboxes below andtotheright of theithboxofthediagonal,readingfromlowerrighttoupperleft.Frobenius cate($10)tehraetrinesofthepartion(Manyweernowweaba. areverse notationforthecharacteristics, writing(b,,...,by|a,,-..,a,)instead.) For thepartition (10,9, 9,4,4,4, 1) (SEERESSE| HiEpEeEE] oA asaharactersios= Ht characteristics (67) CET uy Algebraically, rand thecharacteristics a,<-** <a, and by<-- <h,are deterinined byrequiring theequality ofthetwosets S24. Representations ofG,:Young Diagrams andFrobenivs’s Character Formula Uncle =1e@ggk= Lea} and {0b kW By k+B). (b)Show that ¥(xVots) =oI (9),where y=x—dand [Te- = Jo)=5, at)=fm []5+0 foratyis Deduce thatthesumin(4.18) isthecoefficient ofx~*ing(x\/f(x). {c)When m=2,usethistoprove theformula dim Vs, -ink = ala, xl(12)=GqPy(bb+D~aa,+1) Hurwitz [Hur] used thisformula ofFrobenius tocalculate thenumber of ‘ways towrite agiven permutation asaproduct oftranspositions. From this ‘hegaveaformula forthenumber ofbranched coveringsoftheRiemannsphere with agiven number ofsheets andgiven simple branch points. Ingram [In] hhasgiven other formulas forz,(g), when gisasomewhat more complicated conjugacy class Exercise 4.19*. IfVisthestandard representation ofS,,prove thedecom- positions into irreducible representations: Sym?V =U@V@ Va-2.2» V@V =Sym'V OAV =U BVOMur. ®Merny Exercise 4.20%, Suppose Aissymmetric, ie, A=2’,andletq,>9>">4,>Oethelengthsofthesymmetric hooksthatformthediagramof2;thus,gy=2A, —1,gy=2A,—3,.... Show that ifgisaproduct ofdisjoint cycles oflengths 44,435-1 dr then 2g) =(-De", §4.2. Irreducible Representations ofS, Weshow next that therepresentations V,constructed inthefirst section are exactlytheirreducible representations ofG,.Thisproofappearsinmanystandard texts (eg. [C-R], [Ja-Ke], [N-S], [Wet]), sowewill beatittle concise. LetA=CG, bethegroup ring ofS,.For apartition Aofd,letPand Q bethecorresponding subgroups preserving therowsandcolumns ofaYoung tableau Tcorresponding to4,leta=a,, b=by,and let c=c, =abbe H.2. Irreducible Representations ofGy 3 thecorresponding Young symmetrizer, soV;=Ac, isthecorresponding representation. (These groups andelements should really besubscripted by Ttodenote dependence onthetableau chosen, buttheassertions made depend only onthepartition, soweusually omit reference to.) Note that PQ =(1},soanelement ofGcan bewritten inatmost one wayasaproduct p-4, p€P,q€Q.Thus, cisthesumJ,bey,thesumoveralgthatcanbewrittenasp-4,withcoefficient 41beingsgn(q);inparticular,thecoefficient ofe,inis1. Lemma421.(1)Forp€P,p-a=a°p=a.(2)For qeQ,(sgn(q)q):b =b-(sgn(q)q) =b. )ForallpeP, q€0,p-c-(sen(q)a) =c,and, uptomultiplication bya scalar, etstheonly such element inA. Proor. Only thelastassertion isnotobvious. IFng¢,satisfies thecondition in(3),then iyy=s80(@)n, forallgp,q:inparticular, py=sen(q)n,. Thus, itsuffices toverify that n,=0if@¢PQ.Forsuch gitsufices tofind transposition ¢such that p= Pandq=gg €Q;forthen g=paq,80 1,=~My IT” =gTisthetableau obtained byreplacing each entry iofT byofl), theclaim isthat there isaretwo distinct integers that appear inthe same row ofTandinthesame column of7’;tisthen thetransposition ofthesetwointegers.Wemustverilythatiftherewerenosuchpairofintegers,thenonecouldwriteg=p-qforsome p¢P,q€Q.Todothisfirsttakep,€P and4;€0’=gQg™ sothatp,Tandqi7”have thesame frstrow: repeatingontherestofthetableau,onegetsp€Pandq’€Q’sothatpT=q’T".Then,pT=497,s0p=4q'g,andthereforeg=pq,whereq=9"'(q') taeQ,asrequired, a Weorder partitions lexicographically: A>p_ifthe firstnonvanishing 2~1ispositive (422) Lemma 4.23.(1)1f4>sthenforallx©Ayay-x-b,=0.1mparttenlar,if2 >3 thene,-¢, =0.Q)Forallx€A,eq*x°€, =isascalarmultipleofcy.Inparticular,e,¢y =nea forsome n,€C. Proor.For(I),wemaytakex=g€G,,Sinceg-b,-q"tistheelementcon-structed from 97", where T”isthetableau used toconstruct b,,itsulfices toshowthata,°b,=0.Oneverifiesthat2>jimpliesthattherearetwointegers inthesame row ofTand thesame column of7°.If¢isthetransposition of these integers, then ay:t =dy, ¢-by=—by,$0dy"By=ay°CEby=44"bye asrequired. Part (2)follows from Lemma 42t(3. fal Exercise 424°, Show thatif#thency:A-c, =0:inparticular, ey"¢, =0. 54 4,Representations ofGy:YoungDiagramsandFrobenius’s CharacterFormula Lemma 4.25, (1)Each ¥,isan irreducible representation ofGy. 2)IfAx14then ¥,andV,arenotisomorphic Proof. For (I)note that cj¥yc€e, byLemma 423. IfWoh, isa subrepresentation, then ¢,W iseither Cc, or0.Ifthefirst istrue, then Vz=Ase, W.Otherwise W:Wc A-cW =0, but this implies W=0. Indeed, aprojection from Aonto Wisgiven byright multiplication byan clement g€A with @=g?€W-W =O. This argument also shows that€4¥;#0,ie,thatthenumbern,ofthe previous lemma isnonzero. For(2), wemay assumeA>j.Thenc,V=Ce,#0,butcx,=€x°Ay= 0,s0they cannot beisomorphic A-modules. 0 Lemma 426, Forany A,¢,-€4 =mics, with my=di/dim Ky, Proor. LetFberight multiplication byc,onA.Since Fismultiplication by nn,onVj,andzero onKer(c) thetrace ofFisn,times thedimension ofButthecoefficient ofeineyC1is1,$0trace(F)=|G,=dl. fal Sincethereareasmanyirreduciblerepresentations V,asconjugacyclasses ofG,,these must form acomplete setofisomorphism classes ofirreducible representations, which completes theproofofTheorem4.3.nthenextsection wewillprove Frobenius’s formula forthecharacter ofV,,and, inaseries ofexercises,discussalittleofwhatelseisknownaboutthem:howtodecompose{ensor products orinduced orrestricted representations, how (ofind abasis, TorYa,ete §4.3. ProofofFrobenius’s Formula For any partition 2ofd,wehave asubgroup, often called aYoung subgroup, 6-8, x KG, OG, a2 LetU;betherepresentation ofG,inducedfromthetrivialrepresentation of;,Equivalently, U;=A+a,, with a,asinthepreceding section. Let a= Xo,=character ofUs. (428) Key tothisinvestigation istherelation between U,and Vj,i,between Yj and thecharacter z,of¥;.Note first that V,appears inU,,since there isa surjection Uy=Aay—»Va=Addy,xrx-by (4.29) Alternatively, Vy=Aah, &Abya,©Aa,=Us, 3, ProofofFrobenivw’s Formula 8s byExercise 44,Forexample, wehave Uaca.n®hay®Vay which expresses (hefactthatthepermutation representation C*ofG,isthe sum ofthe standard representation andtherival representation, Eventually ‘wewillseethatevery U;contains ¥;with multiplicity one, andcontains onlyotherVyfor>2‘The character ofU;,iseasy tocompute directly since U,isaninduced representation, andwedothisnext. For i=(iy... 1a)d-upleofnon-negativeintegerswithYai,=d,denote by Gee theconjugacy classconsisting ofelements madeupofi,I-cycles,i,2-cycles,,izd-cycles.Thenumberofelements inC;iseasilycountedtobe dl 1Gimi dag 430) Bytheformula forcharacters ofinduced representations (Exercise 3.19), 1 NG) =Ale: GIGS VIG)=gqlSe: SAFIGOSs1 hited at s a!a Le | dt Tht ZTPeribodral where thesum isover allcollections {tyq:1< p<kt <q<d} ofnon- negative integers satisfying parig ttag to he Ayagt+gytootdg(TocountC,n;,writethepthcomponent ofanelementofG,asaproduct ofrpsL-cycles,r,2-cycles, ....)Simplifying, A i WIG)=>qFaltaoral (431)thesumoverthesamecollectionsofintegers{FyThissumisexactly thecoefficient ofthemonomial X?=x}*-...-xj*in the power sum symmetric polynomial PO (xyeons tag Odto tM le bo xf (4.32) ‘Sowehave the formula VAG)=[P™],=coefficient ofXinPt, (433) Toprove Frobenius’s formula, weneed tocompare these coefficients with the coefficients «»,(i) defined by 564,Representations ofG,:YoungDiagramsandFrobenius’CharacterFormula o,f)=[APM,b=Aytk—Ayth2)...ab (434) Our goal, Frobenius’s formula, istheassertion that x,(C;) =«,(i). “There isageneral identity, valid foranysymmetric polynomial P,relating such coefficients: [PLY Kyl: Pigstetieptt-2s oon where thecoefficients K,,arecertain universally defined integers, calledKostkanumbers. ForanypartitionsAandof,theintegerK,,maybedefined combinatorially asthenumber ofways tofilltheboxesoftheYoungdiagram forjwith A,I's,A;2s,uptoA,K's,insuch away that theentries ineach rowarenondecreasing, andthose ineach column arestrictly increasing; such arecalledsemistandard tableaux on1oftypeA.Inparticular, Kyat, andKy,=Oforp<d ‘The integer K,,may bealso bedefined tobethecoelficient ofthemonomial X=x}'-...-x2* intheSchur polynomial S,corresponding toj.Forthe proof that these areequivalent definitions, see(A.9) and(A.19) ofAppendix ‘A.In thepresent case, applying Lemma A.26tothepolynomial P=P*,we deduce WG)=FKya, =0.0)+LKyo. (439) “The result ofLemma A.28 canbewritten, using (4.30), inthe form 1 GIMME, =dup (436) This indicates that thefunctions w,,regarded asfunctions ontheconjugacy classes ofG,,satisfy thesame orthogonality relations astheirreducible characters ofS,.Infact, onecandeduce formally from these equations that the«must betheirreducible characters ofG,, which iswhat Frobenius proved. Alittle more work isneeded toseethat «,isactually thecharacter ‘oftherepresentation V,that is,oprove Proposition 4.37. Letx= zy,bethecharacter ofV,.Then foranyconjugacy class C,ofSy, 1G) =o,f) Proor. Wehave seen in(4.29) that therepresentation Us,whose characteris a,contains theirreducible representation ¥,.Infact, thisisallthat weneed {oknow about therelation between U,and ¥,.Iimplies that wehave a=Stanza tas2Wallmy20, (438) Consider thisequationtogetherwith(4.35).Wededucefirstthateach«,isa $4.3. ProofofFrobenius's Formula 7 virtual character: wecan write 1=LmytmMpEZ Butthe«like thez,,areorthonormal by(4:36), so 1=(@n0,) =Emin andhencewis+:forsomeirreducible characterz.(Itfollowsfromthehooklength formula that theplus sign holds here, butwedonotneed toassume this) FixA,and assume inductively thatz,=0,forall >2,0by(4.35) Waront FKate Comparing thiswith(4.38),andusingthelinearindependence ofcharacters, theonly possibility isthat3=2, a Corollary 439(Young's rule). Theinteger K,isthe multiplicityoftheirredue- iblerepresentation V,intheinduced representation Uy UZUODK uM, Watt LKate Note that when 2(1,..., 1},Usisjust theregular representation, so Kyu,_.1) =dim V,.This shows that thedimension ofVjisthenumber ofstandardtableauxon2,i.e,thenumberofwaystofilltheYoungdiagramofAwiththenumbersfrom1tod,suchthatallrowsandcolumnsareincreasing,‘Thehooklengthformulagivesanothercombinatorial formulaforthisdimen-sion. Frame, Robinson, and Thrall proved that these two numbers areequal Forashortandpurelycombinatorial proof,see[G-N-W].Foranotherproof thatthedimension of¥,isthenumberofstandardtableaux,see[Jam].Thelatter leads toacanonical decomposition ofthegroup ring A=CS, asthe direct sum ofleftideals Aer, summing over allstandard tableaux, with ey=(dim ¥,/dl)-ey, andcytheYoung symmetrizer corresponding toT,cf.Exercises4.47and4.50.This,inturn,leadstoexplicitealculation ofmatrices‘oftherepresentations ¥,withintegercoefficients.Foranother example ofYoung's rule, wehave adecomposition Yacun=BDManto Infact, theonly jxwhose diagrams canbefilled with d—aI'sand a2's, nondecreasing inrows andstrictly increasing incolumns, arethose with at ‘most two rows, with thesecond row nolonger than a;and such adiagram hasonty onesuch tableau, sothere arenomultiplicities. Exercise 440°. The characters yj,ofG,have been defined only when Aisa partition ofd.Extend thedefinition toanyk-tuplea=(a,,...,a4)ofintegers 584,Representations of&,:YoungDiagramsandFrobenius’sCharacterFormula that add uptod bysetting ¥,=0ifany ofthea,arenegative, and otherwise Wa=WaywhereJisthereordering ofa,,...,a,indescending order.Inthiscase¥,isthecharacter oftherepresentation induced fromthetrivialrepresen- tation bytheinclusion of&,,x--- xS,inSp.Use(A.5) and(A9) ofAppendix A(oprovethedeterminantal formulafortheirreducible characters ain terms oftheinduced characters Y,: a=F800 srtgeeay-2ondrahe Ionewritesy,asaformalproductYo,"---"Vaystheprecedingformula can bewritten Ya dager Vaya =Wasyabe artMe Vago Va Theformalproductofthepreceding exerciseisthecharacter versionofan “outer product” ofrepresentations. Given anynon-negative integers dj,...,dy,andrepresentations ¥jofG,,,denotebyV,0°»0the(isomorphism class.ofthe)representation ofS,,d =Sd),induced fromthetensor product repre-sentation Y,8°:¥,ofG,,x“xG,,bytheinclusionofGy,x="xSy,inS,(see Exercise 2.36). This product iscommutative and associative. Itwill turn out tobeuseful tohaveaprocedure fordecomposing sucharepresenta- tionintoitsirreducible pieces.Forthisitisenoughtodothecaseoftwofactors,andwiththeindividual representations V,irreducible. Inthiscase,onehas,forV,therepresentation ofS,corresponding tothepartition Aofdand ¥,therepresentation of©,corresponding tothepartition j«ofm, V0Vy=NapsMor (441) thesum over allpartitions vofd+m,withN,,,thecoefficients givenbythe Littlewood-Richardsonrule(A.8)ofAppendixA.Indeed,bytheexercise,the characterofVo¥,istheproductofthe corresponding determinants, and, by(A8),thatisthestimofthe characters Nj,Whenm=1andx=(m),V,istrivial;thisgives Ind"Ys=Mos (42) thesumoverallvwhoseYoungdiagramisobtained fromthatofAbyadding ‘onebox. This formula usesonlyasimplerformoftheLittlewood-Richardson ruleknown asPieris formula, which isproved in(A.7). Exercise 443%. Show thattheLittlewood-Richardson number Nyyyisthe multiplicity oftheirreducible representation V,03V, intherestriction of¥, from &,,,, toS,xS,,.Inparticular, taking m=1,4=(I),Pieri’s formula (A.7) gives: Rese"¥, =5Vay $43. ProofofFrobenius's Formula 9 thesum over all4obtained from vbyremoving one box. This isknown asthe“branching theorem,”andisusefulforinductiveproofsandconstructions, particularly because thedecomposition ismultiplicity free. Forexample, you ‘canuseittoreprove thefactthatthemultiplicity ofV,inU,isthenumber of semistandard tableaux onoftype 2.Itcanalso beused toprove theassertion made inExercise 4.6that therepresentations corresponding tohooks areexteriorpowersofthestandardrepresentation. Exercise 4.44* (Pieri’s rule). Regard ,asasubgroup ofS,,,, asusual. Let4 bea partition ofdandvapartition ofd+m.UseExercise4.40toshowthat themultiplicityofV,intheinducedrepresentation Ind(¥,)iszerounlessthe Youngdiagram ofJiscontained inthatofv,andthenitisthenumberof‘waystonumbertheskewdiagramlyingbetweenthemwiththenumbersfrom|tom,increasing inboth row and column. ByFrobenius reciprocity, this is thesame asthemultiplicity of¥,inRes(¥). ‘Whenappliedtod=0(or1),thisimpliesagainthatthedimensionofV,isthe numberofstandardtableauxontheYoungdiagramofv. Forasampling ofthemany applications ofthese rules, see[Dia §7,§8]. Problem 4.45*.TheMurnaghan- Nakayama rulegivesanefficient inductivemethod forcomputing character values: If2isapartition ofd,andg€S,is writtenasaproductofanm-cycleandadisjointpermutation h€G,_,,,then 19) =L(-44(, where thesum isoverall partitions «of d—m that areobtained from 4by removing askewhookoflengthm,andr(ji)isthenumberofverticalstepsin theskew hook, i-e.,onelessthan thenumber ofrows inthehook. Askew hook for2isaconnected region ofboundary boxes foritsYoung diagram such that removing them leaves asmaller Young diagram; there isaone-to-one correspondence between skew hooks andordinary hooks ofthesame size, as indicated: A=(7,6,5,5,4,4,1, 1) =(1,4,4,3,3,1, 151) hook length =9,r=4 Forexample, if4hasnohooksoflengthm,thenx,(g)=0. ‘The Murnaghan -Nakayama rule may bewritten inductively asfollows: If 8awritten asaproduct ofdisjoint cycles oflengths my,my... ny,With thelengths m,taken inanyorder, then z,(9) isthesum(— 17",where the sum isoverall ways stodecompose theYoung diagram of2bysuecessively 60.4,Representations of&,:YoungDiagramsandFrobenivs'sCharacterFormula removing pskew hooks oflengths m,,...,my, andr(3)isthe(otal number of vertical steps inthehooks of. (a)Deduce theMurnaghan—Nakayama rulefrom (4.41) andExercise 4.16, using theLittlewood-Richardson rule. Or: (b)Withthenotation ofExercise 4.40,showthat Vester Wald)=3Vahey Wel Exercise 4.46°. Show that Corollary 4.39 implies the“Snapper conjecture”: theirreducible representation V,occurs intheinduced representation U;if and only if SasSmforallj>t. Problem 447*. There isamore intrinsic construction ofthe irreducible representation V,,calledaSpechtmodule,whichdoesnotinvolveofthe choice ofatableau; itisalso useful forstudying representations ofS,inpositive characteristic. Define atabloid {T} tobeanequivalence class oftableaux(numberings bytheintegers1tod)onA,twobeingequivalent iftherowsare thesame uptoorder. Then G,actsbypermutations onthetabloids, andthe corresponding representation, with basis thetabloids, isisomorphic toUs Foreach tablea 7;define anelement Fin thisrepresentation space, by By=be(T} =Zsenta)(ar}, thesum over theqthat preserve thecolumns ofT.The span ofallE,'s is isomorphic toV,,and theE,'s, where Tvaries over thestandard tableaux, form abass. Another construction ofV,istotake thesubspace ofthepolynomial ring C[x,,....%4] spanned byallpolynomials F,,where F;=[](x;—x),the prodict overall pairsi<Jwhichoceurinthesamecolumnithetableau7: Exercise4.48*.LetU;betherepresentation A-b,,whichistherepresentationof&,induced from thetensor product ofthealternating representations on thesubgroup S,=G,,x“ xG,,, where jt=2istheconjugate partition. Show thatthedecomposition ofUris Ue DK yrke Deduce that V;istheonly irreducible representation that occurs inboth U; andU;,anditoccurs ineach with multiplicity one. Note,however, thatingeneralAc#Asa, A-hysinceAc,maynotbe contained inAa, $43.ProofofFrobenius's Formula 6 Exercise4.49*,Withnotationasin(441),ifU’=Vg...)isthealternatingrepresentation ofG,,show that V,0Y,,..1) decomposes into adirect sum Vz, thesum over allxwhose Young diagram can beobtained from that of Abyadding mboxes, with no(wo inthesame row. Exercise 450, Wehave seen that A=C6, isisomorphic (oadirect sum of1m,copiesofV,=Acy,wherem,=dimV;isthenumberofstandardtableau onA.This canbeseen explicitly asfollows. Foreach standard tableau Ton each 4,letcybetheelement ofCS, constructed from T.Then A=@A cr. Indeed,anargumentlikethatinLemma4.23showsthalcy:cy-=Owhenever Tand T”aretableaux onthesame diagram and T>T’,i.thefirst entry (reading from lefttoright, then (optobottom) where thetableaux differ hastheentryofTlargerthanthatof7”.FromthisitfollowsthatthesumEAcyisdirect. Adimension count concludes theproot. (This also gives another proof that thedimension of¥,isthenumber ofstandard tableaux on2, provided oneverifies that thesum ofthesquares ofthelatter numbers isdt, ef[Boe] or[Ke]) Exercise 451*. There areseveral methods fordecomposing atensor product oftworepresentations of©,which amounts (ofinding thecoefficients Cy, inthedecomposition Vi@ Vy ECan for2,1,and ypartitions ofd.Since oneknows how toexpress Vyinterms oftheinduced representations U,,itsuffices tocompute ¥;@U,, which isisomorphic toInd(Res(V,)), restricting and inducing from thesubgroup &,=©,xS,,x+;thisrestriction andinduction canbecomputed bythe Littlewood-Richardson rule, For d<5,you can work outthese coefficients using only restriction to©,., andPieris formula. (a)Prove thefollowing closed-form formula forthecoefficients, whichshowsinparticular thattheyareindependent oftheorderingofthe subscripts, Anand: 1 Cn=Leo,o,0o,(0,Fagen thesumoveralli=(ig...ia)with Eat, =d,and with «,()=x4(C)and af)=itHI.vig!(b)Show that _ft ituea 1itpex Exercise 452*, Let Ry=R(S,) denote the representation ring, and set R=GeoRy.Theouterproductof(4.41)determines maps Re@Ra Rese 624, Representations ofGy;Young Diagrams andFrobeniu’s Character Formula which makes Rinto.a commutative, graded Z-algebra, Restriction determines maps Ream =R(Syom)+R(S_xSq)=Ry@Rey which defines aco-product 5:R-»RQR.Together,thesemakeRintoa (graded) Hopf algebra. (This assertion implies many oftheformulas wehave proved inthis lecture, aswell assome wehave not) (@)Show that asanalgebra, RX TH Ha where Hyisanindeterminate ofdegree dH, corresponds tothetrivial representation ofS,.Show that theco-prodvet 8isdetermined by 5UH,)=HOHy@Hy+$1@My IEwesetA= Z[Hjy.-sayes]QAgewecanidentifyAywiththe symmetric polynomials ofdegree dink=dvariables. The basic symmetric Polynomials inA,defined inAppendix Atherefore cortespond tovietual representations of (0)Show that, corresponds tothealternating representation U',and HyesUn Siet¥y BretUb (©)Show that thescalar product(,)definedonA,in(A.16)coreesponds (othescalarproductdefinedonclassfunctionsin(2.10)(G)Show that theinvolution 9ofExercise A.32 corresponds totensoring arepresentation with thealternating representation U'. (©)Show that theinverse map from R,toAytakes arepresentation 1to 1shrwlGadP®FaqgaelGo) where26)=i112...hts “The(innertensorproductofrepresentations ofG;gives amap Ry@Ry—> ,which corresponds toan“inner product” onsymmetric functions, some- times denoted « (©)Show that 0 forint 6pmmere{lovitjek Since these P*form abasis forAy@ Q,thisformula determines theinner product LECTURE 5 Representations of%,andGL,(F,) Inthislectureweanalyzetherepresentationoftwomoretypesofgroups:thealternat- inggroupsandthelineargroupsGL(F)andSL,(F,)overGiteelds.ntheformer ‘cate, weprove some general results relating therepresentations of@group tothe fepresentationsofasubgroupofindextwo,and.usewhatweknowaboutthesymmetric group; thisshould becompletely straightforward given just thebasic ideas oftherecedinglecture.InthelatercaewestartessentiallyfromscratchThetwosections«anberead(otnot)independent; neitherislogicallynecessaryfortheremainderofthe book 45.1:Representations of§52: Representations ofGL,(,) and SL(,) §5.1. Representations of%, “The alternating groups &,,d >5,form one oftheinfinite families ofsimple groups. Inthissection, continuing thediscussion of§3.1, wedescribe their irreducible representations, The basic method foranalyzing representationsofW,isbyrestricting therepresentations weknowfrom©. Ingeneral when HisasubgroupofindextwoinagroupG,thereiaclose relationship betweentheirrepresentations. Wewillseethisphenomenon againinLietheory forthesubgroups SO,oftheorthogonal groups O,. LetUand Udenote thetrivial and nontrivial representation ofGobtained from thetwo representations ofG/H. For any representation VofG,let V'=V@U'; thecharacter ofV'isthesame asthe character ofVon elements ofI,buttakes opposite values onelements notinHi.Inparticular, Res’ =Res Vv. “a 5.Representations ofandGL3(F,) IfWis any representationofH,thereisaconjugaterepresentation defined byconjugating byanyelement ¢ofGthat isnotinH;ifyisthecharacter‘ofW,thecharacter oftheconjugate ish++(tht-?). Since¢isuniqueupto multiplication byanelement ofH,theconjugate representation isunique up toisomorphism. Proposition 5.1.LetVbeanirreducible representation ofG,andletW=ResV betherestriction ofVtoH.Then exactly oneofthefollowing holds: (1)Visnocisomorphic toV’;Wisirreducibleand isomorphictitsconjugate; IndgW=VvoV'.(2)V&V;W=W'@W*,whereW'andW"areirreducible andconjugate‘butnotisomorphic; tnd W"xInd W"=V. Each irreducible representation ofHarises uniquely inthisway, noting that incase(1) V'and Vdetermine thesame representation. Proor. Letxbethecharacter ofV.Wehave 16l=2M=¥beer+Ftx0oP- Since thefirst sum isan integral multiple of, this multiple must beIor2, which arethetwo cases oftheproposition. This shows that Wiseither irreducibleorthesumoftwodistinctirreducible representations W"andW”. ‘Note that thesecond case happens when y(t)=0forall¢¢H,which isthe case when V"isisomorphic toV.Inthesecond case, W’and W"must be conjugate since Wissel-conjugate, and ifW'and W”were self-conjugate V wouldnotbeirreducible. Theotherassertions in(1)and(2)followfromtheisomorphism Ind(ResV)=V@(U@U’)ofExercise 3.16.Similarly, forany representation WofH,Res(Ind W)isthe direct sum ofWanditsconjugate— asfollows sayfrom Exercise 3.19—from which thelast statement followsreadily o Mostofthisdiscussion extends withlittlechange tothecasewhereHisanormal subgroup ofarbitrary prime index inG,cf.[B-tD, pp.293-296]. Gifford hasextended muchofthispropositiontoarbitrarynormalsubgroups offinite index, cf.[Dor, §14]. ‘There aretwo types ofconjugacy classes ¢inH:those that arealso conjugacy classes inG,and those such that cUc’ isaconjugacy class inG, where c’=tct™', «¢H;thelatter arecalled split. When Wisirreducible, its character assumes thesamevalues—those ofthecharacter oftherepresenta-tionVofGthatrestrictstoW—onpairsofsplitconjugacy classes,whereasintheother case thecharacters ofW’and W”agree onnonsplit classes, buttheymustdisagreeonsomesplitclasses.Ifzw-(c)=xy-(c’)=x,andyy-(c’)= Aw-(@) =y,weknow thesum x+y,since itis thevalue ofthecharacter of therepresentation Vthat gives risetoW’and W"oncc’. Often theexact values ofxandycanbedetermined fromorthogonality considerations. $5.1. Representations of&, 6s Exercise 5.2*. Show that thenumberofsplitconjugacyclassesisequaltothe number ofirreducible representations VofGthat areisomorphic toV’,or tothenumber ofirreducible representations ofHthatarenotisomorphic {otheirconjugates.Equivalently, thenumberofnonsplitclassesinHissame asthenumberofconjugacy classesofGthatarenotinH. Weapply these considerations tothealternating subgroupofthesymmetric group. Consider restrictions oftherepresentations ¥,from ©toWy.Recall that if’istheconjugate partition toA,then Ye WOU, with U’thealternating representation. The two cases oftheproposition correspond tothecases (I)4”#Aand (2)2°=A.12¥A,letWybethe restriction ofV,(0Wy.If2’=2,letW;and Wybethetworepresentations ‘whose sum istherestriction of¥;.Wehave Indy=Vi@Vy,Resy=ResVy=WywhenX¥A,IndWi=IndWe=H,Resi=Wi@WywhenY=2 Note that #(self-conjugate representations ofS,) =#{symmetric Young diagrams) =#(splitpairsofconjugacy classesinW,} =#(conjugacyclassesin©,breakingintotwoclassesinWy) Nowaconjugacy classofanelementwrittenasaproductofdisjoint cycles issplit ifandonly ifthere isnooddpermutation commuting with it,which is equivalent (0allthecycles having odd length, and notwo eycles having the same length. Sothenumber ofself-conjugate representations isthenumber ofpartitions ofdasasumofdistinctoddnumbers.Infact,thereisanatural correspondence between these two sets: any such partition corresponds to& symmetric Young diagram, assembling hooks asindicated: [EEE |Her|}HtGeHHH nnn a) IfAisthepartition, thelengths ofthecycles inthecorresponding split conjugacy classes areqy=2d—1dy=2Ay~3,43 =2dy—5)... 6 5,Representations of andGL3(F,) Foraselfconjugate partition 4let andxjdenote thecharacters ofWj and W;,andlet cand’ beapairofsplitconjugacyclasses,consistingofeycles ‘ofodd lengths q,>q2><" >q.The following proposition ofFrobenius ‘completes thedescription ofthe character table of Proposition 5.3.(1)Ifcandc'donotcorrespond tothepartition 2,then 140 =2il€) =49 =2) =Sraleve) (2)Ifcandc’correspond toA,then BO=GE=%, le =GO=» with xandythetonumbers (rt VO a), andm=S([]9;—1)=4d=9. Forexample,ifd=4and4=(2,2),wehaver=2,q,=3,q2=1,andx and yarethecube roots ofunity; therepresentations Wjand Wj"arethe representations labeled U’and U"inthetable in§2.3. For d=5,4=(3,1,1), r=1,q;=5, andwefind therepresentations called Yand Zin§3.1. For 4<7, there isatmost onesplit pair, sothecharacter table canbederived from orthogonality alone.Notethatsinceonlyonepairofcharactervaluesisnottakencareofby thefirst case ofFrobenius's formula, thechoiceofwhichrepresentation isW andwhich W,"isequivalent tochoosing theplus andminus sign in(2).Note also that theinteger moccurring in(2)isthenumber ofsquares above the diagonal intheYoung diagram of2. ‘Weoutline aproof oftheproposition asanexercise: Exercise 54°. Step1.Letq-=(qi>°~>qj)beasequenceofpositiveodd imtegers adding tod,and letc’=c’(q) and c"=c*(q) bethecorresponding conjugacy classesinW,.LetAbeaself-conjugate partitionofd,andlet7and3bethecorresponding characters of&,.Assume that7,andxjtakeonthe‘same values oneach element of%,that isnotinc’or¢”.Letw=x3(c') = Aile"Vand v=4(€°) =rile’) (@Showthatvandvarerealwhenm=$5(q,~1)iseven,and7=vwhenmis odd. (ii)Let9.=x,—x3.Deduce from theequation (9,9)=2thatju—0]?= (ii)Show that isthepartition thatcorresponds toqandthat +0= (=1)%, and deduce that «and varethenumbers specified in(2)ofthe proposition, Step 2.Prove theproposition byinduction ond,andforfixed d,look at thatqwhichhassmallest q,,andforwhichsomecharacter hasvaluesontheclasses c’(q) andc*(q) other than those prescribed bytheproposition, $52,Representations ofGLy(F,)andSLs(f,) 0 ()Mfr=t,s0q,=d=2m+1,thecortesponding self-conjugate partition isA=(m+ I,1,..., IByinduction, Step Iapplies toz;andzi. (i)Ifr> 1,consider theimbedding H=W,,xyg, ¢G=%,,andlet X’and X"bethe representations ofGinduced from therepresentations Wie@Wy and Weas, where Wjand W7aretherepresentations ofWf,corresponding toq;,i.e,tothesel-conjugate partition(a,~1},ly...»I)OF4435Wyisoneofthe representations of8, corresponding to(q2,....q,).and{Gdenotestheexternaltensorproduct(seeExercise2.36).ShowthatX°andX"areconjugate representations of@,,and their characters z’and x”take ‘equal values oneach pairofsplitconjugacyclasses,withtheexceptionofea) andc“(q), and compute thevalues ofthese characters onc'(q) and c"(q)- (ii) Let =y’~*,and show that (8,8)=2.Decomposing X'and X" into their irreducible pieces, deduce that X"=¥@Wjand X"=¥® Wjfor some sel-conjugate representation Yandsome sell-conjugate pattition Aof. (iv)ApplyStep1tothecharacters y,andy},andconcludetheproof. Exercise 5.5*. Show that ifd>6, theonly irreducible representations of1,ofdimension lessthan darethetrivial representation and the(1—1)dimensional restriction ofthestandardrepresentation ofG,.Findtheexcep-tions ford <6. WehaveworkedoutthecharactertablesforallG,and%,ford<5.With theformulasofFrobenius,aninterestedreadercanconstructthetablesfor@ fewmored—untilthenumberofpartitions ofdbecomes large. §5.2. Representations ofGL,(F,) andSL,(F,) ThegroupsGL4(F,)ofinvertible2x2matriceswithentriesinthefiniteield F,with qelements, where qisaprime power, form another important series offinite groups, asdotheir subgroups SLa(f,) consisting ofmatrices of determinant one. Thequotient PGL,(F,) =GL,(F,V/FZ_ istheautomor- phism group ofthefinite projective lineP4(F,). Thequotients PSLa(F,)= SLy(F,)/{ 41} aresimple groups ifq#2,3(Exercise 5.9). Inthissection we sketch thecharacter theory ofthese groups. Webegin with G=GL3(F,). There areseveral keysubgroups: ‘ab icool Der-{6 i} (This “Borel subgroup” Band thegroup ofupper triangular unjpotent ‘matricesNwillreappearwhenwelookatLiegroups.)SinceGactstransitively ‘ontheprojective lineP'(F,), withBtheisotropy group ofthepoint (1:0), we have {Gl=|BI-/P'(F I=(q~1a+1) 6 5.Representations ofandGL(F,) Wewillalso need thediagonal subgroup 2") pees,oaf(tSherer. wherewewriteFforFy.Let”=F,:betheextensionofFofdegreetwo,uniqueuptoisomorphism, Wecanidentify GL,(F,) asthegroup ofallF-linearinvertibleendomorphisms ofF.ThismakesevidentalargecyclicsubgroupK=(F" ofGAtleastifgisodd,wemaymakethisisomorphismexplicitb hootingsgenertogforthecelepoupF*andchoosing«squarerootJe inF’.Then Iand«/eformabasisforF'asavector space overF,sowecan make the identification: f(xosye(:)k= =r, etext yes{¢epeeyxseeastive KisacyclicsubgroupofGoforderq?~1.Weoftenmakethisidentification,leavingitasanexercisetomakethenecessarymodifications incaseqiseven.“Theconjugacy classes inGareeasily found: Representative No.ElementsinClassNo.Classes x0) xt >»-(5!) e- ot x0) 2 @=)@-2) -( 2 aa=) or Herec,andc, «areconjugateby(_) }),andd,, andd,..,areconjugatebyany(:<=)TocountthenumberofelementsintheconjugacyclassofbylookattheactionofGonthisclassbyconjugation; theisotropygroup{(6hesthenumberofelements intheclassistheindexofthisgroup inG,whichsq?—1.Similarlytheisotropygroupfor¢,.,isD,andtheisotropygroupford,,isK.Toscethattheclassesaredisjoint,considertheeigenvaluesandtheJordan canonical forms. Since they account for[Glelements, thelist iscomplete There areq?—1conjugacy classes, sowemust findthesame number of irreducible representations. Consider first thepermutation representation of GonP4(F), which hasdimension q+1.Itcontains thetrivial representation; §52.Representations ofGL(,)andSL.(F,) 6 JetVbethecomplementary q-dimensional representation. The values ofthe‘characteryofVonthefourtypesofconjugacy classesarex(a,)=4,x(b,)=0,Uexy)=1,x(da,,)=—1,whichwedisplayasthetable: Vig 0 1 -1 Since (x,2)=1,Visiceducible. For each ofthe q—I characters a:F*+C* ofF*, wehave aone- dimensional representation U,ofGdefined byU,(g)=a(det(g)).Wealso have therepresentations V,=V@ U,.The valves ofthecharacters ofthese representations are Ug ax? aay ax)aty) a(x? ~69") Ves gat ax)a(y) a(x —ey?) Notethatifweidentify(3°)with(=4y/éinF,then xtaey?=aa(2)Normye(C) =C08= The next place tolook forrepresentations isatthose that areinduced{romlargesubgroups. Foreachpaira,ofcharactersofF*,thereisacharacter ofthesubgroup B: B+ BIN =Dm FtxFY408 xCF CF, whichtakes()toa(a)p(d).LetW,gbetherepresentation inducedfrom Bio Gbythis representation; thisisarepresentation ofdimension (G: B]= 4a+1.ByExercise 3.19 itscharacter valuesarefoundtobe:Woe ENA) —AIL) aCIALY)+aA) Weseefrom thisthat Wy Wpay that WU, Vand thatfora#B therepresentation isirreducible. This gives {(q—1)(q—2)more irreducible representations, ofdimension q+1 ‘Comparing with theistofconjugacy classes,weseethatthereare1q(q—1) irreducible characters lefttobefound. Anatural way tofindnew charactersistoinducecharactersfromthecyclicsubgroupK.Forarepresentation OK =(Fy C4, thecharactervaluesofthe induced representation ofdimension [G: K]= q— Late Ind(@): aq— Nel) 9 9 e+ 00K Hereagain (=x +y/eeK =(F')*. Note thatInd(p") =Ind(g), sothe representations Ind(g) for9*#@give }a(q ~1)diferent representations. 70 5.Representations oft,andGL3(F,) However, these represenations arenotirreducible: thecharacter yofInd(o) satisfies (z,x)=q—1if@*#g,and otherwise (z,z)= q.Wewillhave toworkalittlehardertogetitreducible representations fromtheseInd(g)Another attempt tofind more representations istolook inside tensorproductsofrepresentations weknow.Wehave¥,@U,=¥,,andW,4@U,= Wy80therearenonewones(0befoundthisway.Buttensorproductsof theV'sandW,,,'s aremore promising. Forexample, V@W,,,hascharacter values VOM: gat Dax) 0 a(x) tay) 0 Wecancalculatesomeinnerproductsofthese characters with each other toestimate how many itteducible representations each contains, and how many they have incommon, For example, Grom tm.) =2 Ginaen tn =I ifgle=a, Grom trem,)=4 +3, Crem nue) = itgle=a Comparing with theformula (ste) Zinag)) =4~1,one deduces that V@W,, and Ind(9) contain many ofthesame representations. With any luck, Ind(g) andW,,. should both becontained inV@W,..Thisguessis ‘easily confirmed; thevirtual character By=Krom, ~Karas ~Kia takes values (q—a(x), —a(x), 0,and —((C) +o(G)") onthefour types of conjugacy classes. Therefore, (19.15)=1,andx,(I) =q—1>0, s07,is, infact, thecharacter ofanirreducible subrepresentation ofV@W. of dimension q~1.Wedenote thisrepresentation byX,.These ta(q— 1) representations, for ¥@,andwith X,=X,a, therefore complete thelist ofirreducible representations forGL(F) The character table is ' e-! ete ene 20 oa 0) (9) | ay a) ao) ae) Ki] eats?) 0 ac) aie)Was|+NeCIM)—abe) —_alBL)+aL) ° HIa=nets) =9t0) ° tote)+99) Exercise 5.6.Find themultiplicity ofeach irreducible representation inthe representations V@ W,,and Ind(¢). £52, Representations ofGLa(F,)andSL3(,) 1” Exerese 57,Find thecharacter table ofPGL3(F) =GL(FY. Note thaits characters arejust thecharacters ofGL,(F) that take thesame values on elements equivalent mod F Weturn next tothesubgroup SLa(F,) of2x2matrices ofdeterminant one, wth qadd. Theconjugacy lasses, together with thenumberof elementsineachconjugacyclass,andthenumberofeoujugacy classesofeachype,areRepresentative No.ElementsinClass No.Classes 10 0 «-(,‘) \ \ -1 0 -e= 1 f ° .(o-i) vi v=) 9 (') z ' re gt o ()) j -11 g-1 le ee! ° (0-) a ' x0 a-3 om(5Sxeat a+) > xy a-1 ®(5Nxwat aa-1) a Theverifications arevery much aswedidforGLs(F,). In(7),theclases of (Sana(5)arethesame.In(8),theclassesfor(x,y)and(x,—y)arethesame;abefore,betterlabelingisby theelement inthecyclicgroup Cm(Le(Fy crt=1} theelements +1arenotused, and theclasses of(and {-!atethesame. “The total number ofconjugacy classes isq+4, 30weturn tothetask ofFindingq+4irreduciblerepresentations. Wefistseewhatwegetbyrestrct-ingrepresentations from GL,(F,). Sine weknow thecharacters, there ino problem working this out, and sesimply state theresults (1)The Usall restrict tothe trivial representation U.Hence, iwerestrict any representation, wewillgetthesame forall tensor products byUz’. n 5,Representation ofandGL3(F) (2)The restriction Vofthe ¥;8isireducble)Therestriction W,ofW,isireducibleifa? 4I,andW,=W,when=aorf=a", These giveI(q_- 3)iereducible representations ofdimension +1(8)LetedenotethecharacterofF*with«?=Ix1.TherestrictionofM% isthe sum oftwodistine irreducible representations, which wedenote Wand W"(4)TherestrictionofX,dependsonlyontherestrictionoftothesubgroupGvand gand@"!determinethesamerepresentation. Therepresenation isirreducible if@?+1.This gives {(q—1)itreducible representations of dimension q~1.(4)IGdenotesthecharacterofCwithy?=1,4#{therestrictionofXy isthesum oftwodistinct irreducible representations, which wedenote X’and X", Altogether this listgives q+4distinct irreducible representations, and it istherefore thecompete list.Tofinish thecharacter table theproblem isto describe thefour representations W",W", X',and X".Since weknow thesumofthesquaresofthedimensions ofallrepresentations, wecandeducethatthesum ofthesquares ofthese four representations isq?+1, which isonly Possible ifthefits twohave dimension f(q-+ 1)andtheother two4(q~1) ‘This issimilar towhat wesawhappens forrestrictions ofrepresentations to subgroups ofindex two. Although theindex here ilarger, wecanusewhatwweknowaboutindextwosubgroupsbyfindingasubgroup11ofindex wo inGL,@,) thatcontains SL(F,) andanalyzing therestrictions ofthese four representations toH. ForHwetakethematrices inGL(F,) whote determinant isasquare. The representatives oftheconjugacy classes arethesame asthose forGL(F,),including,ofcourse,onlythoserepresentatives whosedeterminant isasquare, butwemastadclesrepresented bytecemene(5),20.These areconjugatetotheelements(;a)inGL,(F,),butnotinH.Thesearethe4~|splitconjugacyclasses,Theprocedureoftheprecedingsectioncambeusedtoworkoutalltherepresentations ofH,butweneedonlyaliteofthisNote that thesign representation U'from G/HT isU,,s0 that W,, = 1,@U'andX,=X,@ U'stheir restrictions toHspit intosums ofconju: aie irreducible representations ofhalf their dimensions. This shows these representations stayirreducible onrestriction from HHtoSL(F,)s0 thatIV"and1"areconjugaterepresentations ofdimension4(q+1),andX’andX*areconjugate representations ofdimension {(q— 1).Inaddition, weknow that thelr character values onallnonsplit conjugacy clases arethesame ashafthecharactersoftherepresentations W,andX,.respectively. Thisialtheinformation weneed tofinish thecharacter table. Indeed, theonly values notcovered bythisdiscussion are {52 Reyrniaion ofGLandSL, a Vt) (le) (tt) (te OL OL 0=1, o-! woos ot . ¢ ‘The first two rows are determined asfollows. We know that 5+(= tt 1oiyt_(i-t (},|)taiscongrent tomodulo4,andto(5$)otherwise,andsine x0")=7a)foranycharacter,weconcludethatsand¢arerealifq= 1mod(4), ands=Tifq=3mod(4).Inaddition,since—eactsastheidentity ‘orminus theidentity foranyirreducible representation (Schur’s lemma), x(a) =Xa) x(t) forany irreducible character x.This gives therelations s’=c(—1)s and 1=¢(—1)t. Finally, applying theequation (x,z)=1tothecharacter ofW’ gives@formula forsi+(3.Solving theseequations givess,1=$+}./wa,where =¢(—1) isor—laccording asq=1or3mod(4). Similarly one ‘computes that wand vare—}+ 4,/«g. This concludes thecomputations needed towrite out thecharacter table. Exercise 5.8.Byconsidering theaction ofSL,(F,)onthesetP'(F,),showthat SL,(F,) &S5,PSL,(F,) &W,,andSL_(F,) =Us. Exercise 5.9%,Usethecharacter table forSL,(F,) toshow thatPSL,(F,) isa simple group ifqisodd and greater than 3. Eercbe 10 Comput thecharacter able ofPSL) te byrearing itasaquotient ofSL,(F,), orasasubgroup ofindex twoinPGL,(E,). Exercise 5.11*. Find theconjugacy classesofGL,(F,),andcomputethechar- actersofthepermutation representations obtainedbytheactionofGL3(F,) ‘on(j) theprojective plane P?(F,)and(ii)the“flagvariety”consisting ofapoint ‘onaine inP2(E,). Show thatthefirstisirreducible andthatthesecond isa som ofthe evil recent, wo cope heit representation, bd anirteducible representation. Athogh techaracters ofheabove groups were found bytheeaty pioneers inrepresentation theory, actually producing therepresentations in natural way ismore difficult. There hasbeen agreat dealofwork extending ” 5.Representations ofWand GLa(F,) thisstory toGL,(F,) andSL,(F,) forn>2(cl.[Gr]}, andforcorresponding, groups, called finite Chevalley groups, related toother Liegroups. Forsome hints inthisdirectionsee[Hu3),aswellas[Ti2].Sinceallbutafinitenumber offinite simple groups arenow known toarise this way (orarecyclic or alternating groups, whose characters wealready know), such representations play afundamental role ingroup theory. Inrecent work their Lie-theoretic origins have been exploited toproduce their representations, buttotellthis story would gofarbeyond thescope ofthese lecture(®)s. LECTURE 6 Weyl’s Construction Inthislectureweintroduceandstudyanimportantcollectionoffunctorsgeneralizingthesymmetric powers andexteiot powers. These aedefined sinply interms ofthe Young.symmettizersitrodoed inpiven representation Vofan arbitrary Brou? G,weconsider thedthtensor power ofV,onwhich both Gandthesymmetric groupcondlettersact.Wethentaketheimageoftheactionofc,onV®;thisisagainarepresentation ofG,denoted,(V).Thisgivesusawayofgenerating newrepresenta-tion,whosemainapplicationwilbetoLiegroups:forexample,wewillgeneratallreprerentations ofS,Cbyapplyingthesetothestandardrepresentation C*ofSLC. While itmay beeasiest toread thismaterial while thedefinitions oftheYours symmettizers arestilrsh inthemind, theconstruction wilnotbeused again until $15, sothat thislecture canbedeferred until then. 46.1:Schurfunctorsandtheircharacters$6.2: The proofs §6.1. Schur Functors and Their Characters Foranyfinite-dimensional complex vector space V,wehave thecanonical decomposition VOV~SymV@NV. ‘ThegroupGL(V)actsonV®V,andthisis,asweshallsoonsee,thedecom- positionofV®Vintoadirectsumofirreducible GL(V)-representations. For thenext tensor power, V@V@V =Sym*V OAV®another space. ‘Weshall seethat thisother space isasum oftwocopies ofanirreducible %6 6Weyl's Construction GL(V}representation, JustasSym‘V andMVareimages ofsymmetrizing operators from V& =V@ V@-~" Vtoitself,soaretheotherfactors.The symmetric group ©,actsonV®,sayontheright, bypermuting thefactors (6,@04):0=t9qPa This action commutes with theleftaction ofGL(V). Foranypartition 2ofdwehave from thelastlecture aYoung symmetrizer ¢,in€G,. Wedenote theimageof¢,onV®bySV: S,V =Im(cshes) which isagain arepresentation ofGL(V). Wecallthefunctor! V~'S,¥ the ‘Schur functor orWeyl module, orsimply Weyl’s construction, corresponding toA.ItwasSchur who made thecorrespondence between representations of symmetric groups andrepresentationsofgenerallineargroups,andWeylwho made theconstruction wegivehere.” Wewillgiveother descriptions later, cf. Exercise6.14and§155. Forexample, thepartition d=dcorresponds tothefunctor V~~Sym* V, andthepartition d=1++" +1tothefunctor V~M¥. Wefind something new forthepartition 3=2+1.The corresponding syminetrizer is Caan 14ean~fas~Eusay 0theimage ofc, isthesubspace ofV®spanned byallvectors 0,BOs +02@oyW035 —vO, Ot —By WP ILA'V.@ Visembedded in¥® bymapping (1A03)@04+0,O02@v3—OMB,thentheimageofcisthesubspaceof\?V@Vspannedbyallvectors (01405) 02+(0A09)@01. Itisnothard toverify thatthese vectors span thekernel ofthecanonical mapfromA?V@VtoAV,sowehave Sa.nW =KeriV@ VAY). (This gives themissing factor inthedecomposition ofV®) Note that some oftheS,canbezero ifVhassmall dimension. Wewill seethatthisisthecase precisely when thenumber ofrows intheYoungdiagramofAisgreaterthanthedimension ofV. "Thefuncoraity meansslythatnearmapg:V+Wfvectorspacesdetermines nearsnap 5419) 51¥ +SW, with $,(9 6)~Sy(¢)» SH) and$d) =Id 2Thenotiongoesbyavarityofnamesandnoatinsintheerature,dependingonthecomes Constructionsfermarkedlywhennoloverafieldofcharactetitc zero;ndmanyathorsnow rarametize them bytheconjeate partion Out choice ofnotation fsguided hythecorre: spondencebetwentheeupctors andScrpolynotins, whichwewiseaetheichacacters $6.1. Schur Functors and Their Characters nn When G=GL(V), andforimportant subgroups G<GL(V), these S,Vgivemanyoftheirreducible representations ofG;wewillcomebacktothis Tater inthebook. For now wecanuseourknowledge ofsymmetric group reptesentations toprove afewfacts about them— inparticular, weshow that they decompose thetensor powers V%, andthat they areirreducible repre- sentations ofGL(V).Wewillalsocomputetheircharacters; thiswilleventually beseen tobeaspecial case oftheWeyl character formula. ‘AnyendomorphismgofVgivesrisetoanendomorphism of§,V:Inorder toellwhat representations weget, wewillneed tocompute thetrace ofthis endomorphismonS,V;wedenotethistraceby75,y(9).Forthecomputation, letx,,.-., %betheeigenvaluesofgonV,k=dimV.Twocasesareeasy.For reid SV =Sym! tap =Halsys 0% ) where H,(x,,..., Xs)isthecomplete symmetric polynomial ofdegree d.The defition ofthese symmetric polynomials given in(A.1) ofAppendix A ‘Thetruthof(6.1) isevident when gisadiagonal matrix, and itstruth forthe dense setofdiagonalizable endomorphisms implies itforallendomorphisms; fofone canseeitdirectly byusing theJordan canonical form ofg.For 2(Lec Dywehave simitasly ScoanYMMAP=Eleiert —(62) with B40 ..00%4) theelementary symmetric polynomial [see (A.2)]. The polynomials H,and E,arespecial cases oftheSchur polynomials, which wedenoteby5,=8431,4)ABvariesoverthepartitionsofdintoameet keparts, these polynomials §,form abasis forthesymmettic polynomials of degree dinthese kvariables. Schur polynomials aredefined anddiscussed in ‘Appendix A,especially (A.4)-(A.6). The above twoformulas canbewritten A510) =S,0y.005%4) forA=(A)andA=(yes De Wewillshow that thisequation isvalid forall2: ‘Theorem 63. (1)Let k=dim V.Then S,V iszero ifAyn #0. If2= > 2A205then Anasini dim $,V =S,(1,..., I)= poieBatSane ™ CoD WL Fa (2)Letm,bethedimension oftheirreducible representation V,ofS, corresponding to. Then ya @5,0, (3)Foranyg€GL(V), thetraceofgonS,VisthevalueoftheSchur polmomtalontheelgennalues<u.omaf908 Asie) =Sil6 100) " 6,Weyts Construction (4)Bach§,¥1sanirreducible representation ofGL(V). ‘This theorem willbeproved inthenext section. Other formulas forthe dimension of§,V aregiven inExercises A.30 and A.31. The following is another: Exercise 64°. Show that an Kia pe eet. dim5.7=FETs 49 where theproducts areover thedpairs (i,/)thatnumber therowandcolumnofboxesfor4,andhyisthehooknumberofthecorresponding box. Exercise 6.5,Show that¥®?xSym*V ®A?V @(Sy2,1)¥)%, and VO4aSym*VOMY (Sp, ©Sa2,nV)9? B(S2.4.0V) ‘Compute thedimensions ofeachoftheirreducible factors. ‘The proofofthetheoremactuallygivesthefollowing corollary: Corollary 6.6.If¢€CS,, and(CS,)-c = VE" asrepresentations ofGy, then there isacorresponding decomposition ofGL(V)-spaces: vee PSve If4, 25X4aretheeigenvalues ofanendomorphism ofV,thetrace ofthe induced endomorphism ofVcisS.74Sy(Xy 004%) IfAandpare different partitions, each with atmost k=dim parts, the irreducible GL{V)-spaces §,V and§,V arenotisomorphic. Indeed, their characters aretheSchur polynomials'S, and,,which aredifferent. Moregenerally, atleastforthoserepresentations ofGL(V) which canbedecom- posed intoadirect sum ofcopies oftherepresenations S,V's,therepresenta- tions arecompletely determined bytheir characters. This follows immediately from thefact thatthe Schur polynomials arelinearly independent.Note,however,thatwecannothopetogetallinite-dimensional irreduciblerepresentations ofGL(V) thisway, since theduals ofthese representations arenotincluded. Wewill seeinLecture 15that this isessentially theonly ‘omission. Note alsothatalthough theoperation that takes representations of E,torepresentations ofGL(V) preserves direct sums, thesituation with respect toother linear algebra constructions such astensor products ismore complicated. One important application ofCorollary 6.6istothedecomposition ofa tensor product $,¥ @S,V oftwoWeyl modules, with, say,4apartition of 46.1. Schur Functors and Thei Characters ~ dand 1apartition ofm.Theresultis SV OS,V =DN S.Vs ) here thesum isover partitions vofd+m,andN,,,arenumbersdetermined bytheLittlewoodRichardsonrule.ThisisarulethatgivesN,,,asthenumber ofways toexpand theYoung diagram ofA,using inanappropriate way, to achieve theYoung diagram forv;see(A.8) fortheprecise formula. Two important specialcasesareeasiertouseandprovesincetheyinvolveonlythesimpler Pieri formula (A.7). Forpt=(m),wehave SV@Sym"V= @DS,¥, (68) thesum over allywhose Young diagram isobtained byadding mboxes 10 theYoung diagram ofJ,with notwo inthesame column. Similarly for (lyons De S,VONV=@S,¥, (69) thesumover allpartitions xwhose Young diagram isobtained from that of Aby adding mboxes, with notwo inthesame row. Toprove these formulas, weneed only observe that SV OSV=VOC BVOM cy =VQ VOM (e,@e)=VO, with c=c,@c,€ CS, @CS, =C(S, xS,) <CS,,,,. This proves that S,V@ S,V hasadecomposition asinCorollary 6.6,andthecoefficients are given byknowing thedecomposition ofthecorresponding character. The character ofatensorproductistheproductofthecharactersofthefactors; sothisamounts towriting theproduct 5,5,ofSchur polynomials asalinear‘combination ofSchurpolynomials. ThisisdoneinAppendix A,andformulas 67), (6.8), and(6.9) follow from (A.8}, (A.7), and Exercise A.32 (v),respectively. Forexample, rom Sym!V @V=Sym'*"V ®Sy,,¥,itfollows that Su.1)V =Ker(Sym’V @V+Sym**"V), andsimilarly fortheconjugate partition, SeaatanY =Ke(hV@ V+MV), Exercise 6.10%, One canalso derive thepreceding decompositions oftensor products directly from corresponding decompositions ofrepresentations of symmetric groups. Show that, infact,5 @S,V corresponds tothe“innerproduct”representation V;©V,ofG,,,,described in(4.41), Exercise 6.11*.(a)TheLittlewood- Richardson rulealsocomesintothede-composition ofaSchurfunctorofadirectsumofvectorspacesVandW.This 0 6.West's Construction generalizes thewell-known identities Sym"(V.® W)=BD(Symev@Sym"iv), AVOW)= BWV@NW). Prove thegeneral decomposition over GL(V) xGL(W): SVOW) =DMplS.VOS,W), thesum aver allpartitions 2,such that thesum oftheoumbers partitioned byA.and sisthenumber partitioned byv.(Tobeconsistent with Exercise 2.36 one should usethenotation &forthese “external” tensor products)(6)SimilarlyprovetheformulafortheSchurfunctorofatensorproduct: S(VOW) =DClS.VOS,0), wherethecoefficients C,,,aredefinedinExercise4.51.Inparticularshowthat Sym(V@ W)= OSV OS", thesum overall partitionsofdwithatmostdimVordimWrows.Replacing WbyIV*,thisgivesthedecomposition forthespaceofpolynomialfunctions cofdegree don thespace Hom(¥, W)over GL(V) xGL(W), Forvariations conthistheme, see[Ho3]. Similacly, NV@W)=DS,V OS", thesum over partitions 2ofdwithatmostdimVrowsandatmostdimW columns. Exercise 6.12. Regarding GL,C=GL,€ x{1}GLCxGL,CoGy thepreceding exercise shows how therestriction ofarepresentation de- composes: Res(S,(C"™™)) =E(Naye dim§,(€"))5(C°). Inparticular, form=1,Piersformulagives Res(S.("')) =BSC), thesum over all2obtained from vbyremoving anynumber ofboxes from itsYoung dingram, with notwo inanycolumn. Exercise 613°. Show that foranypartition 1=(sty...) ofdy NVORVO-ONVE ORySeh, where K,,istheKostka number and 2'theconjugate of2 §6.1. Schur Functors andTheitCharacters ar Exercise 6.14*. Letji=2’betheconjugate partition. Putthefactors ofthedthtensorpowerV®inone-to-one correspondence withthesquaresofthe‘Youngdiagramof2.Showthat§,istheimageofthiscomposite map: QUNV) +Bi(G"V) +VOB(G4)-»Wy(Sym), thefirst map being thetensor product oftheobvious inclusions, thesecond‘grouping thefactorsof¥®according tothecolumnsoftheYoungdiagram,thethird grouping thefactors according totherows oftheYoung diagram, andthefourth theobvious quotient map. Alternatively, S,Vistheimage of acomposite map @ym*V) -»@(@*V) +VE+@(OV)+@,(WV). Inparticular, S,V canberealized asasubspace oftensors inV®thatare invariant byautomorphisms that preserve therows ofaYoung tableau of4,orasubspace thatisanti-invariant underthosethatpreserve thecolumns, butnotboth, ef.Exercise 4.48. Problem6.15*.Theprecedingexercisecanbeusedtodescribeabasisforthespace 5,V. Letr,.-.,1% beabasis forV.For each semistandard tableau 7(on2,onecanuseittowritedownanelementvyin(AV); vyisatensorproductofwedgeproducts ofbasiselements, theithfactorinA*Vbeingthe ‘wedge product (inorder) ofthose basis vectors whose indices occur intheith column of1:The fact tobeproved isthat theimages ofthese elements vyunderthefirstcomposite mapofthepreceding exerciseformabasisforS,V. AttheendofLecture 15,using more representation theory than wehave atthemoment, wewillwork outasimple variation oftheconstruction of SV which willgive quick proofs ofrefinements ofthepreceding exercise and problem. Exercise 6.16*. The Pieri formula gives adecomposition Sym!V@Sym'V=DSusa-n¥s thesum over 0<a <d.The lefi-hand side decomposes into adirect sum of ‘Sym?(Sym*V) andA?(Sym*V), Show that, infact, Sym*(Sym*V) =S24,0)V ®Si24-2,2V ®Si24-4oV Os ASyMV) =Sass.nY BSas-s.a1¥ @Sears. O° ‘Similarly using thedual form ofPieri todecompose 4V@A‘Vintothesum @Sp¥,thesumoverall2=(2,...,2,1,-..5H)consisting ofd—a2'sand2a V's,0 <a<d,show that Sym?(/“V) isthesum ofthose factors with aeven,audA*(\¢V)isthesumofthosewithaodd, 2 6Wey!s Construction Exercise6.17".Zandareanypartitions, wecanformthecompositefunctor S,(S,V). Theoriginal “plethysm” problem—which remains very difficult in sgeneral—is todecompose these composites: SASAY) =BMay, thesumoverallpartitions vofdm,where2isapartitionofdandjispartition ofm.Thepreceding exercisecarriedoutfourspecialcasesofthis. (a)Show that there always exists such adecomposition forsome non- negative integers M,,, byconstructing anelement cinCS,q, depending on Aand 1,suchthat§,(5,¥) isVc. (b)Compute Sym?(S,,, .,¥)andA*(S,2,2)¥). Exercise 6.18* “Hermite reciprocity.” Show that ifdim V=2there areiso- morphisms Sym*(Sym*V)=Sym*(Sym?V) ‘ofGL{V}-representations forallpand g. Exercise619°.MuchofthestoryaboutYoungdiagramsandrepresentationsofsymmetric and general linear groups canbegeneralized toskew Young diagrams, which atethediflerences oftwo Young diagrams. Ifand y.ate partitions with 4<2, foralli,4/jtdenotes thecomplement oftheYoung. diagram foryinthat ofA.For example, ifA=(3,3,1)and w= (2,1),A/nis thenumbered part of ‘Tocach A/s:wehave askew Schur function S,)4,which canbedefined byanyofseveralgeneralizations ofconstructions ofordinary Schurfunctions.Using thenotation ofAppendix A,thefollowing definitions areequivalent: 0) Sie =Macny-tsh «i Sup=NBxi-n-tash i Sug=Emaxooo88%wherem,isthenumberofwaystonumbertheboxesof2/1 with a,I's,a;2's, ++4k's,with nondecreasing rows and strictly increasing columns. Interms ofordinary Schur polynomials, wehave ) Sun=ELNanaSes where N,,,istheLittlewood-Richardson number. $6.1, Schur Functors and Their Characters 3 Each A/udetermines elements ayq, byy.,aNdYoung symmetrizers Cay=yb,inA=CS,,d=¥2,—py,exactlyasin§4.1,andhencearepresenta- tiondenoted Vy,=Acyy, ofSj.Equivalently, V,,,istheimage ofthemap Abjy-+ Ady given byright multiplication bya,,,,ortheimage ofthemap Ady, -+Abs), given byright multiplication byb,,,.Thedecomposition ofVijy intoirreducible representations is 7) Van=ENaraYe Similarly there areskew Schur functors Sy, which takeavector spacetotheimageofc,,,onV;equivalently, S,,,Vistheimageofanaturalmap (generalizing that intheExercise 6.14) w QUNHV) +VOB Sym'r®V), i) QiSym'nV) +VOB(MAW). Given abasis v,,.... forVand astandard tableau Ton 2/u, one canwrite down anelement vyin@,(A'%*V); forexample, corresponding tothedis- played tableau, vy=v4@0;Ov, Avs). Akeyfact, generalizing theresult of Exercise 6.15, isthat theimages ofthese elements under themap (vi)form a basis forS,),V. Thecharacter ofS,,,V isgiven bytheSchur function Sy: ifgisan endomorphism ofVwith eigenvalues x,, ...,x4,then wip Hssgr(O)=Supl¥4y0X4)IntermsofbasicSchurfunctors, co) Sun¥=NaSV. Exercise 6.20*. (a)Show thatif4=(p,4),Sip.isthekernelofthecontrac tion map cg:SYMPV@Sym*V-»Sym?*V.®@Syme"Y, (b)If.=(p, 4,7), show thatS,.,.¥ istheintersection ofthekernels oftwocontraction maps¢,_@I,andI,®c,,,,whereI,denotestheidentitymap ‘onSym! ¥. Ingeneral,for2=(A,,...,Ay),SV¢Sym"V@---@Sym'*¥ istheinter- section ofthekernels ofthek—1maps WAN O- Oly, Oly Bly, OoShy Usisk= (©)ForA=(p,1,...., 1),showthatS,Visthekernelofthecontraction map: Serato ¥=Ker(Sym’¥ @APY—»Sym"'V @APY) Ingeneral, forany choice ofabetween 1and k— 1,theintersection of “4 6,Weyts Construction the kernels ofally,except #,isS,V@S,V, where a=(A,,...,2,) and T=ysis Aah 805,¥ isthe kernel of@contraction map defined on S.V@§,V. Forexample, ifaisk —I,and wesetr=24,Pier'sformulawrites S,V@ Sym'V asadirect sum ofS,V and other factors SV; thegeneral assertion in(b)isequivalent totheclaimthatS,Vistheonlyfactorthatis inthekernel ofthecontraction, ie., SAV =Ker(Sqy,..14-y¥ @Sym'V+VN @Sym'-"V). ‘These results correspond towritingtherepresentations V,<U,ofthesym- metric group astheintersection ofkernels ofmaps toUs... ay+t,ajoy-teooede Exercise 6.21, Thefunctorial nature ofWeyl's construction hasmany conse-quences, whicharenotexplored inthisbook.Forexample, ifE,isacomplexofvector spaces, thetensor product E%isalsoacomplex, andthesymmetric ‘group &,actsonit;when factorsinE,andE,aretransposed pasteachother, theusual sign(—1)"isinserted. Theimage oftheYoung symmetrizer c,isa complex S,(E,), sometimes called aSchur complex. Show that ifE,isthe complex E_,=V+Eg=¥,with theboundary map theidentity map, and A=(d), then §,(E,)istheKoszulcomplex ONANQS!NFBS? VNQS+S190, where \'=A’'V,andS!=Sym’¥. §6.2. The Proofs Weneed first asmall piece ofthegeneral story about semisimple algebras, which wework outbyhand. Forthemoment Gcanbeany finite group, although ourapplication isforthesymmetric group. IfUisaright module over A= C6, let B=Homg{(U, U)=(9:U+U:@(0-9) =9(v)-g, WoeU,g eG}. NotethatBactsonUontheleft, commuting with theright action ofA;Biscalledthecommutator algebra.IfU=()U2"isanirreducibledecomposition with U;nonisomorphic irreducible right A-modules, then bySchur’s Lemma. 17 B=BH,Home U™,UE") =BiMalOh, whereM,(C)istheringofn,xm,complexmatrices. IfWis anyletA-module, thetensor product U@,W=U@¢Wsubspace generated by{a@w—v@aw} isaleftB-module byactingonthefirstfactor:b-(v®w)=(b-2)®w. $62. The Proof 85 Lemma 622. LetUbeafinite-dimensional right A-module. (i)Foranyc€A,thecanonical mapU@,Ac-+Ucisanisomorphism ofleft B-modules (i)IfW= Acisanirreducible leftA-module, then U@,W=Ucisan irreducible leftBemodule, Gii)IfW,= Ac;arethedistinct irreducible leftA-modules, with m,the dimension ofWW,then U=DAU®W)™=Di(Uey™™ isthedecomposition ofUintoirreducible leftB-modules. Proor. Note first that Acisadirect summand ofAasaleftA-module; this is ‘aconsequence ofthesemisimplicity ofallrepresentations ofG(Proposition 1.5).Toprove (i),consider thecommutative diagram UQA —+ UG Ac >URA i wheretheverticalmapsarethemaps»@a¥-+v-a;sincethelefthorizontalmapsaresurjective, therightonesinjective, andtheoutsideverticalmapsareisomorphisms, themiddle vertical map must beanisomorphism, For (ii),consider first thecase where Uisanirreducible A-module, so B=C.ItsufficestoshowthatdimU@,W<|.ForthisweuseProposition329. identify Awithadirect sum(Py-, My, ofrmatrix algebras. WecanidentityWwithaminimalletidealofA.Anyminimalidealinthesumofmatrix algebras isisomorphic toone which consists ofr-tuples ofmatrices which arezero except inonefactor, and inthisfactor areallzero except for ‘onecolumn. Similarly, Ucanbeidentified with theright ideal of-tuples which arezero except inonefactor, andinthatfactor allarezero except inonerow. Then U@,Wwillbezero unless thefactor isthesame forUand W,inwhichcase U@,Wcanbeidentified with thematrices which arezero except inonerowandcolumnofthatfactor.ThiscompletestheproofwhenUisirreducible. Forthegeneralcaseof(i,decompose U=(),U9"intoasumofirreducible right A-modules,soU@,W=B,(U@,WIP=COforsomek,whichis visibly irreducible overB= M,,(C). art(i)follows, sincetheisomorphism A2(EW; determines aniso- morphism UZUQAzU® (QW) =PiU@M)™. oO Toprove Theorem 6,3,wewillapply Lemma 6.22totherightCS,-module U=¥®.That lemma shows how todecompose UasaB-module, where B 86 6.Weyls Construction isthealgebra ofallendomorphisms ofUthat commute with allpermuta- tions ofthe factors. Theendomorphisms ofUinduced byendomorphisms of Varecertainly inthis algebra B.Although Bisgenerally much larger than End(V), wehave Lemma 6.23. Thealgebra Bisspanned asalinear subspace ofEnd(V®) by End(V). AsubspaceofV®isasub-B-module ifandonlyifiisinvariantby Guy), Proor. Note thatifWisanyfinite-dimensional vector space, thenSymW is, thesubspace ofW® spanned byallw!=dlw@---@w aswruns through W.Applying thistoW=End(V) =V*@Vproves thefirststatement, sinceEnd(V®)=(V*)™@V%=WE,withcompatible actionsofG,.Thesecondfollows from thefactthatGL(V) isdense inEnd(V/). o ‘Weturn now totheproof ofTheorem 6.3.Note that $V isUc, s0parts (2)and (4)follow from Lemmas 6.22 and 6.23. Weusethesame methods to ive arather indirect butshort proof ofpart (3);foradirect approach seeExercise6.28.FromLemma6.22wehaveanisomorphism ofGL(V)-modules: SV xVey (624) with Vy=Ase, Similarly forUy=Asay, andsince theimage ofright multi-plicationbya,onV®isthetensorproductofsymmetric powers,wehave Sym*V@SymV@---@Sym*V =VE@,Us. (6.28) Butwehaveanisomorphism U,&@),Kya,ofA-modules byYoung'srule(4.39), s0wededuce anisomorphism ofGL(V}modules Sym41V.@ Sym2V.@-@Sym*V =DKyS,V. (6.26) Bywhat wesaw before thestatement ofthetheorem, thetrace ofgontheleft-handsideof(6.26)istheproductH,(x,,..., 4)ofthecompletesymmetricpolynomials H,(x1,...,%) Let5,(g) denote theendomorphism of$,V determined byanendomorphism gofV.Wetherefore have Higley, 4)=EpKyaTraee(S,(g)). Butthese areprecisely therelations between thefunctions H,andtheSchur polynomials 5,[see formula (A.9)J, and these relations areinvertible, since thematrix K,,)ofcoefficientistriangularwithI'sonthediagonal.Ifollows thatTrace(Sy(g)) =Sy(xy,...X4) which proves part (3). Note that if2=(Ay,...5 4) with d>Kanddys, #0, thissame argument shows thatthetrace is5,(X1,.-.*4,0,--s0), which iszero, forexample by (A) Forgtheidentity thisshows that$V =0inthiscase. From part(3) wealso get {62 The Proofs "7 dim $,V=S,(1,..., (6.27)andcomputing 5,(1,...,1)viaExerciseA.30(i)yieldspart(1). o Exercise 6.28. Ifyouhave given anindependent proof ofProblem 6.15, part (3)ofTheorem 6.3canbeseen directly. Thebasis elements vyforS,V specified inProblem 6.15areeigenvectors foradiagonalmatrixwithentriesx1,...,.%as with eigenvalue X*=xf'-...: xf*, where thetableau Thasa,1's,a,2's,..., a,k's.Thetrace istherefore JK, X*,where K,,isthenumber ofways to number theboxes oftheYoung diagram of4with a,I's,a,2's, ...,.a, k's.This isjust theexpression for5,obtained inExercise A.31(a). ‘We conclude this lecture with afew ofthe standard elaborations ofthese ideas, inexercise form; they arenotneeded inthese lectures. Exercise 629°. Show that, inthecontext ofLemma 6:2, ifUisa faithfulA-module,thenAithecommutator ofits commutator B A=(i:U+Us(bv)=by(o),oeU,beB}. IUisnot faithful, thecanonical map from Atoitsbicommutator issurjective. Conclude that, inTheorem 6.3,thealgebra ofendomorphisms ofV®that commute with GL(V)isspannedbythepermutations inSy Exercise 6.30. Show that, inLemma 6.22, there isanatural one-to-one cor- respondence between theirreducible right A-modules U,that occur inUand theirreducibleeftB-modulesFShowthatthereisacanonicaldecomposition U=@Miee v) asaleftB-module andasaright A-module. This shows again thatthenumberofimes¥,occursinUisthedimensionofU,andduallythatthenumberoftimes U,occurs isthedimension of¥;.Deduce thecanonical decomposition ve=DSV WcMy thesumoverpartitions 4ofdintoatmostk=dimVparts;thisdecomposition iscompatible with theactionsofGL(V)and&,.Inparticular,thenumberof times V,occurs intherepresentation V®ofG;isthedimension ofS,V. Exercise 6.31. Letebeanidempotent inthegroup algebra A=CG,andlet U=eA bethecorresponding right A-module. LetF=ede, asubalgebra of A.Thealgebra structureinAmakeseAaleftE-module. Showthatthisdefines anisomorphism ofC-algebras B= ede=Hom(ed, eA)=Homg(U, U)=B 8 6Weyls Construction Exercise 6.32, I1isasubgroup ofG,and ¢¢CH isanidempotent, corre- sponding oarepresentation W=CHeof H,show thatCG eistheinduced representation Indj(W). Forexample, if9:11 +C*isaone-dimensional representation, then Log, Ind9(9)=CG-es,wherees=YHiale (9) 91G)2,Haier PART II LIE GROUPS AND LIE ALGEBRAS From anaive point ofview, Liegroups seem tostand attheopposite endofthespectrumofgroups from finite ones." Ontheonehand, asabstract groups they Seem enormously complicated: forexample, being ofuncountable order, there isnoquestion ofgiving generators andrelations. Ontheother hand, they docome with theadditional data ofatopology and amanifold structure; thismakes itpossible—and, given theapparent hopelessness ofapproaching them purely asalgebraic objects, necessary—to usegeometric concepts to study them, Liegroups thus represent aconfluence ofalgebra, topology, andgeometry, which perhaps accounts inpart fortheir ubiquity inmodern mathematics. It alsomakes thesubject apotentially intimidating one: tohave tounderstand, both individually andcollectively, allthese aspects ofasingleobjectmaybe somewhat daunting. Happily,justbecausethealgebraandthegeometry/topotogy ofLiegroup aresocloselyentwined,thereisanobjectwecanusetoapproachthestudy ‘ofLiegroupsthatextractsmuchofthestructureofaLiegroup(primarily itsalgebraic structure) while seemingly getting ridofthetopological com- plexity. Thisis, ofcourse, theLiealgebra. The Liealgebra is,atleast according {oitsdefinition, apurely algebraic object, consisting simply ofavectorspace with bilinear operation; and soitmight appear that inassociating toaLie sroup itsLiealgebra wearenecessarily giving upalotofinformation about thegroup. This is,infact, notthecase: asweshall sceinmany cases (andperhapsallofthemostimportant ones),encodedinthealgebraicstructureofALiealgebraisalmostallofthegeometryofthegroup.Inparticular, wewill "tnspite ofthithere aredeep only partial understood, elation: betnen finite and Le soups extending even totheir simple group casiteaions 0 ILieGroups and LieAlgebras seebytheendofLecture 8that there isavery close relationship betweenrepresentations oftheLiegroupwestartwithandrepresentations ofthe Lie algebra weassociate 10it;and bytheend ofthebook wewill make that correspondence exact. Wesaid that passing from theLiegroup toitsLiealgebra represents a simplification because iteliminates whatever nontrivial topological structure thegroup may have had; it“attens out,” or“linearizes,” thegroup. This, in turn, allows forafurther simplification: since aLiealgebra isjust avectorspacewithbilinearoperation, itmakesperfectsense,ifweareaskedtostudyareal Liealgebra (oroneover anysubfield of€)totensor with thecomplex numbers. Thus, wemay investigate firstthestructure andrepresentations of complex Liealgebras, and then goback toapply this knowledge tothestudy ofreal ones. Infact, thisturns outtobeafeasible approach, inevery respect: thestructure ofcomplex Liealgebras tends tobesubstantially simpler thanthatofreal Liealgebras; and knowing therepresentationsofthecomplexLie algebrawillsolvetheproblemofclassifying therepresentations ofthe realone. ‘There isone further reduction tobemade: some very elementary Liealgebratheoryallowsustonarrowourfocusfurthertothestudyofsemisimple Liealgebras. This isasubset ofLiealgebras analogous tosimple groups in that they areinsome sense atomic objects, butbetter behaved inanumberofways:semisimple Liealgebraisadirectsumofsimple ones; there areeasy criteria forthesemisimplicity ofagiven Liealgebra; and, most ofall,their representation theory canbeapproached inacompletely uniform manner. Moreover, asinthecase offinite groups, there isacomplete classification theorem forsimple Liealgebras. Wemay thus describe ourapproach totherepresentation theory ofLiegroupsbythesequenceofobjects Liegroup ~~ Liealgebra ~~complex Liealgebra -~semisimple complex Liealgebra. Wedescribe thisprogression inLectures 7-9. InLectures 7and 8weintro-ducethedefinitions ofandsomebasicfactsaboutLiegroupsandLiealgebras. Lecture8ends with adescription oftheexponential map, which allows usto establish theclose connection between thefirsttwoobjects above. Wethen do,inLecture 9,theveryelementary classification theory ofLiealgebras that ‘motivates ourfocus onsemisimple complex Liealgebras, andatleast state theclassification theorem forthese. This establishes thefactthat thesecond, third, and fourth objects above have essentially thesame irreducible repre- sentations. (This lecture may also serve togive abrief taste ofsome general theory, which ismostly postponed tolater lectures orappendices) InLecture 10wediscuss examples ofLiealgebras inlowdimensions. I,LieGroups andLieAlgebras 3 From that point onwewillproceed todevote ourselves almost exclusively tothestudy ofsemisimple complex Liealgebras and their representations. ‘Wedothis, wehave tosay,inanextremely inefficient manner: westart with acouple ofvery special cases, which occupy usforthree lectures (11-13),enuinciatethegeneralparadigminLecture14;carrythisoutfortheclassicalLiealgebras inLectures 15-20; and(finaly) finish offthegeneral theory inLectures21-26.Thus,itwillnotbeuntiltheendthatwegobackandusetheknowledge wehave gained tosaysomething about theoriginal problem. In view ofthislong interlude, itisperhaps agood idea toenunciate onemore time our basic, Point ofView: The primary objects ofinterest areLiegroups and their representations; these arewhat actually occur inreallifeandthese arewhat‘wewanttounderstand, ThenotionofacomplexLiealgebrasisintroducedprimarilyasatoolinthisstudy;itisanessentialtool?andweshouldconsider ‘ourselvesincrediblyluckytohavesuchawonderfully electiveone;butinthe nd itisforus@means toanend. ‘ThespecialcasesworkedoutinLectures11-13aretheLiealgebrasofSLandSL3. Remarkably, most ofthestructure shared byallsemisimple Lie algebras canbeseen inthese examples, Weshould probably point outthat ‘much ofwhat wedobyhand inthese cases could bededuced from theWeyl construction weSawinLecture 6(aswewilldogenerally inLecture 15),but ‘wemainly ignore this,inorder towork from a“Lie algebra” point ofview andmotivate thegeneral story. 2Peshaps notlopkally 80;ef:Adams’ book [Ad]. LECTURE 7 LieGroups InthislectureweintroducethedefinitionsandbasiexamplesofLiegroupsandLiealgebras.Weassumeherefamiliaritywiththedefinitionofdiflerentible manifoldsandraps between them, but nomore; inpartiular, wedonot mention vector fields diflrentiat forms, Riemannian metres, orany other tenors. Section 7.3, which discussesmapsofLiegroupsthatarecoveringspacemapsoftheunderyingmanifolds, ‘maybeskimmedandreferredbacktoasneeded,thoughworkingthroughitwillhelp romote familiarity with basic examplesofLiegroups.74:Liegroups:definitions47.2: Examples ofLiegroups §73: Two constructions §7.1. LieGroups: Definitions You probably already know what aLiegroup is;itisjust asetendowed simultaneously with thecompatible structures ofagroup anda” manifold. “Compatible” here means that themultiplication andinverse operations in thegroup structure x:Gx646 and 1636 areactualy differentiable maps (logically, thisisequivalent tothesingle requirement that themap GxG+ Gsending(x,y)tox:y""is©), ‘Amap, otmorphism, between two Liegroups Gand Hisjust amap :GH that isboth differentiable and agroup homomorphism. Ingeneral,‘qualifiers appliedtoLiegroupsrefertooneoranotherofthetwostructures, 4 7.LieGroups usually without much ambiguity; thus, abelian refers tothegroup structure, n-dimensional otconnected refers tothe manifold structure. Sometimes a condition onone structure turns out tobeequivalent toacondition ontheother;forexample,wewillSeebelowthattosaythatamapofconnected LieBroups:G+Hisasurjectivemapofgroupsisequivalent tosayingthatthedifferential dgissurjective atevery point.‘Oneareawherethereissomepotentialconfusion isinthedefinition ofaLiesubgroup. This isessentially adifficulty inherited directly from manifold theory, where wehave tomake adistinction between aclosed submanifold of amanifold M,bywhich wemean asubset X<M that inherits amanifold structure from M(ie, that may begiven, locally inM,bysetting asubset of thelocal coordinates equal tozero), andanimmersed submanifold, bywhich ‘wemean theimage ofamanifoldXunderaone-to-onemapwithinjective differential everywhere—that is,amap that isanembedding locally inX. ‘Thedistinction isnecessary simply because theunderlying topological space structure ofanimmersed submanifold may notagree with thetopological structure induced bytheinclusion ofXinM.Forexample,themapfromX toMcould betheimmersion ofanopen interval inRintotheplane R?asa figure“6”: Another standard example ofthis, which isalso anexample inthecategoryofgroups,wouldbetotakeMtobethetwo-dimensional realtorusR?/Z?=5"x$!,andXtheimage inMofalineVcRhavingirrationalslope: Theupshotofthis isthat wedefineaLiesubgroup(orclosedLiesubgroup, ifwewant toemphasize thepoint) ofaLiegroup Gtobeasubset that is §7.2.ExamplesofLieGroups 9s simultaneously asubgroup and aclosed submanifold; and wedefine an immersed subgroup tobetheimage ofaLiegroup Hunder aninjective morphism toG.(That aone-to-one morphism ofLiegroups haseverywhere injective differential willfollow from discussions later inthislecture) The definition ofacomplex Liegroup isexactly analogous, thewords “differentiable manifold” being replaced by“complex manifold” and all ‘elated notions revised accordingly. Similarly todefineanalgebraicgroupone. replaces “differentiable manifold” by“algebraic variety” and“differentiable map” by“regular morphism.” Aswewill see, thecategory ofcomplex Lie ‘groups isinmany ways markedly diflerent from that ofreal Liegroups (for‘xainple,therearemanyfewercomplexLiegroupsthanrealones).Ofcourse, thestudy ofalgebraic groups ingeneral isquite different from either ofthesesinceanalgebraicgroupcomeswithafieldofdefinitionthatmayormaynotbeasubfield of€(itmay, forthat matter, have positive characteristic). Inpractice,thoyigh,whilethetwoarenotthesame(wewillseeexamplesofthisinLecture 10,forexample), thecategory ofalgebraic groups over €behavesverymuchlikethecategoryofcomplex Liegroups. §7.2. Examples ofLieGroups ‘ThebasicexampleofaLiegroupisofcoursethegenerallineargroupGL,R.ofinvertible nxnrealmatrices; thisisanopen subset ofthevector space of allnxnmatrices, and gets itsmanifold structure accordingly (sothat the entries ofthematrix arecoordinates onGL,R). That themultiplication map GL,R xGL,R +GL,R isdifferentiable isclear; that theinverse map GL,R +GL,.Ris follows from Cramer's formula fortheinverse. Occasionally GL,R willcome tousasthegroup ofautomorphisms ofann-dimensionalrealvectorspaceV;whenwewanttothinkofGL,Rinthisway(eg,withoutchoosingabasisforVandtherebyidentifyingGwiththegroupofmatrices), wwewillwrite itasGL(V) orAut(V). Arepresentation ofaLiegroup G,of course, samorphism from GtoGL{V). Most other Liegroups aredefined initially assubgroups ofGL, (though theymay appearinothercontextsassubgroupsofothergenerallineargroups, whichis,ofcourse,thesubjectmatteroftheselectures).Forthemostpart, such subgroups may bedescribed either byequations ontheentries ofan1xnmatrix,orasthesubgroupofautomorphisms ofV=Rpreservingsome structureonR®.Forexample,wehave: thespecial linear group SL,8 ofautomorphisms ofR"preserving the volume element; equivalently, mxmmatrices Awith determinant 1 thegroup B,ofupper-triangular matrices; equivalently, thesubgroup of‘tutomorphisms ofR*preserving thelag! “ingenesaflagxxquenceubapacesoffiedvectorspaceeachproperlyconnedin thenexitsxcompleteagcachhasonedimensionlargerthanthepreceding.andportalcberwite, 96 7.LieGroups Deh che hic har, whereKisthespanofthestandardbasisvectors¢,,...y€)-Note thatchoosing,‘adifferent basis and correspondingly adifferent flagyields adifferent sub- group ofGL,R, butoneisomorphic to(indeed, conjugate to)By.Somewhat more generally, foranysequence ofpositive integers a... with sum nwecanlookatthegroupofblock-upper-triangular matrices;thisisthesubgroup‘ofautomorphisms ofR"preserving &partialflag0=KEKemer eyehaRy where thedimension ofWis, + +--+a,Ifthe subspace Fisspanned bythefiesta,+"+a,basisvectorsthegroupwillbethesetofmatricesoftheforin eLefepe\ da Off |tes ofole te olotole/ ja ‘ThegroupN,ofupper-triangular unipotentmatrices(thatis,uppertriangularwith1'sonthediagonal);equivalently, thesubgroupofautomorphisms ofRY preserving thecomplete flag{¥;}where ¥isthespan ofthestandard basisvectorse,,...,€,and actingastheidentityonthesuccessivequotients¥.1/¥;Asbefore, wecan, foranysequence ofpositive integers a...awithsum1, look atthegroup ofbiock-upper-triangular unipotent matrices; thisisthe subgroup ofautomorphisms ofR*preserving apartial lag and acting as theidentity onsuccessive quotients, ie,matrices oftheform TLefepe\ jarofrfete|teofoli ts olololr! ja Next, therearethesubgroupsofGL,Rdefinedasthegroupoftransforma- tions ofV=R"ofdcterininant {preservingsomebilinearformQ:VxV-+¥. IfthebilinearformQissymmetricandpositivedefinite,thegroupweget iscalled the(special) orthogonal group SO,8 (sometimes written SO(nk see p.100). IfQissymmetric andnondegenerate butnotdefinite—e-g, ifithask positive eigenvaluesandInegative—the groupisdenotedSQ,,,RorSOK,I note that SO(K, 1)=SO(, k).IfQisskew-symmetric andnondegenerate, the ‘group iscalled thesymplectic group anddenoted Sp,R; note that inthiscase must beeven ‘Theequations thatdefinethesubgroupofGLRpreservingabilinearform Qareeasytowritedown.IfwerepresentQbyamatrixM—thatis,wewrite Q(o, w)=o Mow $7.2. Examples ofLieGroups 7 forall r,w€R*—then thecondition QlAv, Aw) =Qlo, w) translates into the condition that "oA MAW =foMow forallvandw;thisisequivalent tosaying that “MA =M. Thus, forexample, ifQisthesymmetric form Q(v, w)='o-w given bytheidentitymatrixM=1,thegroupSO,RisustthegroupofxnealmatricesAofdeterminant 1such that ‘A= A“. Exercise 7.1%. Show that inthecase ofSpz,R therequirement that thetransformations havedeterminant |isredundant; whereasinthecaseofSO,R.ifwedonotrequirethetransformations tohavedeterminant |thegroupweget(denoted O,R, orsometimes O(n) isdisconnected. Exercise 7.2%. Show thatSO(k; !)has twoconnected components ifkandare both positive. The connected component containing theidentity isoften denoted SO*(k, 1).(Composing with aprojection onto REorRF,wemay associate toanautomorphism A€SO(k, 1)automorphisms ofRtandRt;80"(k,1)willconsistofthoseA€SOI)whoseassociated automorphismspreserve theorientations ofRand Rt!) Note thatiftheform Qisdegenerate, atransformation preserving Qwill carry itskernel Ker(Q) ={veV:Q(0, »)=0¥we V} intoitself;50thatthegroupwegetissimplythegroupofmatricespreserving thesubspace Ker(Q) and preserving theinduced nondegenerate form Gon the quotient ¥/Ker(Q). Likewise, ifQisageneral bilinear form, that is,neither symmetric norskew-symmetric, alinear transformation preserving Qwillpreservethesymmetric andskew-symmetric partsofQindividually, sowejustgetanintersection ofthesubgroups encountered already. Atany rate, we usually Himit ourattention tonondegenerate forms thatareeither symmetric orskew-symmetric, Ofcourse, thegroup GLC ofcomplex linear automorphismsofacomplex vectorspaceV=C*canbeviewedassubgroupofthegenerallineargroup,GLB; iti, thus, arealLiegroup aswell asisthesubgroup SL,C ofnxm ‘complex matrices ofdeterminant 1.Similarly, thesubgroups SO,C <SL,CandSp2qC SL24Coftransformations ofacomplexVectorspacepreserving,‘asymmetric and skew-symmetric nondegenerate bilinear form, respectively,arerealaswellascomplexLiesubgroups. Notethatsinceallnondegeneratebilinear symmetric forms onacomplex veetor space areisomorphic (inpartic- 98 1.LieGroups ular, there isnosuch thing asasignature), there isonly one complexorthogonal subgroupSO,€cSL,Cuptoconjugation; therearenoanalogsofthegroups SO,,R.AnotherexamplewecancomeupwithhereistheunitarygroupU,orU(r), defined tobethegroupofcomplexlinearautomorphisms ofann-dimensional ‘complex vector space VpreservingapositivedefiniteHermitianinnerproduct HonV.(A Hermitian form Hisrequired tobeconjugate linear inthe first? factor, andlinear inthesecond: H(2o, ow)=ZH(o, w)t, and Hw, 0)= Hi,whitispositivedefiniteif1,e)>Oforv#0) Justasinthecaseofthesubgroups SOandSp,itiseasytowritedowntheequations forU(r):forsomemxnmatrixMwecanwritetheformHas H(G,W)="0°Mow,Yo,weCt{notethatforHtobeconjugate symmetric, Mmustbeconjugate symmetric,ie,M=M);then thegroup U(n)is justthegroup ofnxncomplexmatrices Asatistying WM A=M. Inparticular, if11isthe“standard” Hermitian inner product H(c, w)=‘dwgivenbytheidentitymatrix,U(n)willbethegroupofnxncomplexmatrices Asuch that A= A, Exercise 7.3.Show thatifHisaHermitian form onacomplex vector space V,then thereal part R=Re(H) ofHisasymmetricformontheunderlying real space, and theimaginary part C= Im(H) isaskew-symmetric real form; these arerelated byC(x,w)=R(i, w).Both RandCareinvariant by ‘multiplication byi:R(iv, iv)=R(o, w)Show conversely that anysuch real symmetric Risthereal part ofaunique Hermitian H.Show that ifHis standard,sisRandCcoespods tothematinJ=(1. Dede that. Ult) =OM) Sp2-R. Notethatthedeterminant ofaunitarymatrixcanbeanycomplexnumber ‘ofmodulus 1;thespecial unitary group, SU(n), isthesubgroup ofU(n) of automorphisms with determinant 1.The subgroup ofGL,C preserving an indefinite Hermitian inner product with kpositive eigenvalues and[negativeonesisdenotedU,orU(k,thesubgroupofthoseofdeterminant Iisdenoted SU,,, 0FSU(k, 0.Inasimilarvein,thegroupGLHofquaternionic linearautomorphismsofann-dimensional vectorspaceVovertheringHofquaternions isareal 2Thischoiceofwhichfactorsnearandwhichconjvatneaisestcommonthantheothermakestediferenceinwhatoliowsbutitdoeshavethesmalladvantageofbeingcompatiblewihthenarachoiceforqoaternions. $7.2. ExamplesofLieGroups ” Liesubgroup ofthegroup GL4,R, asarethefurther subgroups ofH-linear transformations ofVpreservingabilinearform.SinceHisnotcommutative, ‘are must betaken with theconventions here, and itsnay beworth alithe digression togothrough this now. Wetake thevector spaces Vtoberight Hemodules; H"isthespace ofcolumn vectors with right multiplication by scalars. Inthisway thenxnmatrices with entries inHactin theusual wayonHi"ontheleft.Scalarmultiplication onthelft(only)isH-linear.View H=C@jC=C?,Thenleftmultiplication byelementsofHtgive Cinear endomorphisms ofC2,which determines amapping H-»MzC to the2x2complex matrices. Inparticular, H*=GLH GsGLC. SimilarlyHt=€7@jC"=C%,sowehaveanembedding GL,HGsGL,,C.Notethat4C-linear mapping g:H"-+ H"isH-linear exactly when itcommutes with £:(e)=(Oj,Iov,+Jo,then0:j=—¥,+J0,,$0multiplicationby jtakes(')t0(° —!)(®)sctotiowsthatiJisthematrixofthepreceding, o) "ro Moy exercise, then GLH =(AeGL,,C: A=A). Thosematriceswithrealdeterminant 1formasubgroupSLyH.‘AHermitianform{or“symplectic scalarproduct’)onaquaternionic vectorspace VisanR-bilineas formK:VxV+Hthatisconjugate H-linear inthe fistfactor andH-linear inthesecond:K(vA,wa)=AK(e,w)y,andsatisfies K(w,0) =K(G,w). ispositive definite ifK(0, 0)>Ofor v¥0.(The conjugate Tofaquaternion A=a+bi+of+dkisdefinedtobea—bi—oj—dk)The standard Hermitian form onH"isZi,w. The group ofautomorphisms ofan ‘dimensional quaternionic space preserving such aform iscalled thecompact symplectic group anddenoted Sp(n)orUy).ThestandardHermitianformon Weis Zim. Exercise 7.4,Regarding Vasacomplex vector space, show thatevery quater- nionie Hermitian form Khas the form K(o, »)=H(0, »)+JQ, »), whereHisacomplexHermitian formand@isaskew-symmetric complexlinearformonV,withHandQrelatedbyQ(c,w)=H(uj,w),andHfsatisfyingthecondition Hj, w))=H(®,w).Conversely,anysuchHermitianHisthe complex part ofauniqueK.IfKisstandard,sois11,and@isgivenbythe same matrix asinExercise 73.Deduce that Spin) =URNASp,,€.Thisshowsthatthetwonotionsof“symplectic” arecompatible. Moregenerally,iKisnotpositivedefinite,buthassignature(p,q)saythestandard Sf-,tm,—Di$,, Hj,theautomorphisms preserving itform a s10up UyqH.Orifthe form isaskew Hermitian form (satisfying thesame 100 2.LieGroups linearity conditions, butwithK(w,0)=—K{(o,w)),thegroupisdenotedumn. Exercise 75. Identify, among allthereal Liegroups described above, which ‘ones are compact. Complex LieGroups Sofar,allofourexamples have been examples ofreal Liegroups. Asfor complex Liegroups,thesearefewerinnumber.ThegenerallineargroupGL,C isone, andagain, alltheelementary examples come tousassubgroups ofthe general linear group GL,C. There is,forexample, thesubgroup SO,C of automorphisms ofann-dimensional complex vector space Vhaving deter- tminant 1and preserving anondegenerate symmetric bilinear form Q(note that Qnolonger hasasignature), andlikewise thesubgroup Sp,C oftrans- formations ofdeterminant Ipreserving askew-symmetric bilinear form. Exercise 7.6.Show that thesubgroup SU(n) ¢SL,C isnotacomplex Liesubgroup.(Itisnotenoughtoobservethatthedefiningequationsgivenabovearenotholomorphic) Exercise 7.7.Show that none ofthecomplex Liegroups described above is compact. We should remark here that both ofthese exercises are immediate con- sequences ofthegeneral factthat anycompact complex Liegroup tsabelian;wewillprovethisinthenextlecture.Arepresentation ofacomplexLiegroup Gisamap ofcomplex Liegroups from GtoGL(V)=GL,Cforann- dimensional complex vector space ;note thatsuch amap isrequired tobe complex analytic. Remarks onNotation A.common convention istouseanotation without subscripts ormention of ‘round field todenote thereal groups: Of), SOM, SO(P.g, Ul, SUL, SU(P.9) Spin) andtousesubscripts forthealgebraic groups GL,, SL, SOy, andSp,. This,ofcourse,introduces someanomalies: forexample,SO,RisSO(n),butSp,isnotSp(n); butsome violation ofsymmetry seems inevitable inanynotation. ‘The notations GL(n, R)orGL(n, C)areoften used inplace ofourGL,R or GL,C, andsimilarly forSL,SO,andSp. Also, where wehave written Sp, some write Sp, Inpractice, itseems that those most interested inalgebraic groups orLiealgebras usetheformer notation, andthose interested incompact groups thelatter. Other common notations areU*(2n) inplace ofourGL,t, Sp(p. 4)forourU,,qH, and O*Q2n) forour UstE. $13, Two Constructions 11 Exercise 7.8.Find thedimensions ofthevarious realLiegroups GL,R, SLR, ByNyySO,R, SO,,,R, Spa, Uln), SU(m), GL,C, SLC, GL,H, andSpin) introduced above §7.3. Two Constructions ‘There aretwo constructions, insome sense inverse toone another, that arise frequently indealing with Liegroups (and that also provide uswith further‘examples ofLiegroups).Theyareexpressed inthefollowing twostatements. Proposition 7.9.LetGbeaLiegroup, Haconnected manifold, and@:H+Ga covering space map.’ Lete'beanelement lying overtheidentity eofG.ThenthereisauniqueLiegroupstructureonHsuchthate”istheidentityand¢is‘map ofLiegroups; andthekernel of@isin thecenter ofH. Proposition 7.10. LetHbeaLiegroup, andTc Z(H) adiscrete subgroup of itscenter. Then there isaunique Liegroup structure onthequotient group G=H/T such that thequotient map H-+GisaLiegroupmap. Theproofofthesecondproposition isstraightforward. Toprovethefirst,one showsthatthemultiplication onGlifts uniquely toanap Hx H+Hwhich takes (e’,¢’)toe’,andverifies that thisproduct satisfies thegroup axioms. In fact,itsuffices todothiswhen Histheuniversal covering ofG,foronecan then apply thesecond proposition tointermediate coverings. o Exercise 7.11%. (a)Show that anydiscrete normal subgroup ofaconnectedLiegroupGisinthecenterZ(G)(b)1F2(G) isdiserete, show that G/Z(G) hastrivial center. ‘These two propositions motivate adefinition: wesaythat aLiegroup map between twoLiegroups Gand Hisanisogeny iftisacovering space map oftheunderlying manifolds; andwesaytwoLiegroupsGandHareisogenousifthere isanisogeny between them (ineither direction). Isogeny isnotan equivalence relation, butgenerates one; observe that every isogeny equiv- lence class hasaninitial member (that is,one that maps toevery other one ‘byanisogeny)—that is,justtheuniversal covering space Gofanyone—and, ifthecenter ofthis universal cover isdiscrete, aswill bethe case forallour semisimple groups, afinalobject G/Z(G) aswell.Foranygroup Ginsuchan ‘equivalence class, wewllallGthesimply connected formofthe group G,and G/Z(G)(iitexists)theadjointform(wewillseelateramoregeneraldefinitionofadjointform). This meane that is«contnoous map wth theropery tha every point ofGhasxreihborhood Uschthat@"(U)«deatunionofopentesexchmappinghomeomor:‘Phically toU, 102 7.LieGroups Exercise 7.12 I11-+Gisacovering ofconnected Liegroups, show thatZ(G) isdiscrete ifandonly ifZ(H) isdiscrete, andthen 11/Z(H) =G/Z(G). There- fore, ifZ(G) isdiscrete theadjoint form ofGexistsandisG/Z(G). ‘Toapply these ideas tosome ofthe examples discussed, note thatthecenterofSL,(overRot€)isjustthesubgroupofmultiplesofthe identity byannth root ofunity; thequotient may bedenoted PSL, orPSL,C. Inthecomplex‘ease,PSL,Cisisomorphic tothequotientofGL,CbyitscenterC*ofscalar‘matrices,andsooneoftenwritesPGL,CinsteadofPSL.C.ThecenterofthegroupSO,isthesubgroup {+1}whenniseven,andtrivialwhen1isodd;in theformer case thequotient willbedenoted PSO,R orPSO,C. Finally thecenterofthe group Spa, issimilarly thesubgroup {1}, andthequotient is denoted PSp,,R orPSPC. Exercise 7.13*, Realize PGL,C asamatrix group, ic,find anembedding. (aithful representation) PGL,C +GLyC forsome N.Dothesame forthe other quotients above. Intheother direction, whenever wehave aLiegroup that isnotsimplyconnected, weeanaskwhatitsuniversalcoveringspaceis,Thisis,forexample,hhow thefamous spin groups arise: aswewillse,theorthogonal groups SO,R. and SO,C have fundamental group Z/2, and sobytheabove there exist connected, two-sheeted covers ofthese groups. These aredenoted Spin, and Spin,C, and willbediscussed inLecture 20;forthetime being, thereader may Finditworthwhile (iffrustrating) totrytorealize these asmatrix groups. The lastexercises ofthis section sketch afewsteps inthis direction which can be done now byhand. Exercise 7.14. Show that theuniversal covering ofU(n) can beidentified with thesubgroup oftheproduct U(n) xRconsisting ofpairs (g,1)with det(g) =e Exercise 7.15. We have seen inExercise 7.4 that SU(2)=Sp(2)=(gH:a9=1}. Identifying R®with theimaginary quaternions (with basis i,j,k), show that, forqj=|,themap1»qugmaps R®toitself, andisanisometry. Verify that theresulting map SUQ) =Spe) +003) isa2:1covering map. Since theequation 9=1describes a3-sphere, SU(2) istheuniversal covering ofSO(3); and SO() istheadjoint form ofSUQ) Exercise 7.16, LetM,C =€*bethespace of2x2-matrices,withsymmetric form Q(A, B)=|Trace(AB"), where BFistheadjoint ofthematrix B;the $73. Two Constructions 103, quadratic form associated toQisthedeterminant. ForgandhinSL3C, themappingAt-+9Ah"isinSOC.Showthatthisgivesa2:|covering SLC xSL{C+SO,C, which, since SLC issimply connected, realizes theuniversal covering of SO,€. Exercise 7.17. Identify C?with thespace oftraceless matrices inM,C, so €SL,C actsbyAr+gAg"!. Show thatthisgives a2:1covering SL,C-+805C, which realizes theuniversal covering ofSOsC. LECTURE 8 LieAlgebras and LieGroups Inthis erucal lectureweintoducethedefinitionoftheLiealgebraassociatedtoaLie 4groupandtselationtothatgroup.Allthresectionsarelogicallynecessaryforwhat follows; $8.1 essential, Weusehere alitle more manifold theory. specially, theAierntalofamapofmanifoldsiusedinafundamental wayin68.1thenotionofthetangent vector toanareinamanifolds used in§8.2 and §8.3, andthe notion ofa‘ctorfieldisintroducedinanauxiliarycapacityin§83.TheCampbellHausdortformulaisintroducedonlytoestabishtheFirstandSecondPrincipleof81below; ifyouarewillingtotkethoseonfaiththeformula(andexercisesdealingwitht)can beskimmed. Exercises 827-820 give alternative descriptions oftheLiealgebra Aansociated to Liegroup, butcanbe skipped fornow. 8.1: Liealgebras: motivation and definition §82 ExamplesofLiealgebras $83:Theexponentialmap §8.1. LieAlgebras: Motivation and Definition Given thatwewant tostudy therepresentations ofaLiegroup, howdowe g0about it?Aswehave said, thenotions ofgenerators andrelations ishardly relevant here. The answer, ofcourse, isthat wehave touse thecontinuous structure ofthegroup.Thefirststepindoingthisis Exercise 8.1.LetGbeaconnected Liegroup, andU<Ganyneighborhood ofthe identity. Show that Ugenerates 6 ‘This statement implies that any map p:G+ HTbetween connected Lie groups willbedetermined bywhat itdoes onanyopen setcontaining the {$81LieAlgebras:Motivation andDefinition 10s identity inG,ie,pisdetermined byitsgerm ate€G.Infact, wecanextend thisidea agood bitfurther: later inthislecture wewillestablish the First Principle: LetGand HbeLiegroups, with Gconnected. Amap p:G+ H isuniquely determined byitsdifferential dp,: T,G+T,Hattheidentity. Thisis, ofcourse, great news: wecancompletely describe ahomomorphism ofLiegroups bygiving alinear map between twovector spaces. Itsnotreally worth that much, however, unless wecan give atleast some answer tothe next, obvious question: which maps between these twovector spaces actuallyariseasdifferentials ofgrouphomomorphisms? TheanswertothisisexpressedintheSecond Principle below, butitwilltake usafewpages togetthere. To start, wehave toaskourselves what itmeans foramap tobeahomomor- phism, and inwhat ways this may bereflected inthedifferential ‘Tobegin with, thedefinition ofahomomorphism issimply a@*map p such that ah) =pa): ath) forallgandhinG.Toexpressthisinamoreconfusingway,wecansaythat4homomorphism respects theaction ofagroup onitself byleftorright multi- plication: thatis,oranyg€Gwedenote bym,:G—+ Gthedifferentiable map iven bymultiplication byg,and observe that a@*map p:G— HofLie groups willbeahomomorphism ifitcarries m,{0my inthesense thatthe diagram G—++H GH ‘commutes. ‘Theproblem withthischaracterization isthat,since themaps m,havenofixedpointsitishardtoassociatetothemanyoperationonthetangentspace toGatonepoint. Thissuggests looking, notatthediffeomorphisms m,,but attheautomorphisms ofGgiven byconjugation. Explicitly, foranyg€Gwe define themap Yy646 by Yh) =g-heg (%,isactually aLiegroup map, butthat isnotrelevant forourpresent Putposes,) Itisnow equally thecase thatahomomorphism prespectstheaction ofagroupGonitselfbyconjugation: thatis,itwillcarry¥,intoVyinthesense that thediagram 106 8.LieAlgebrasandLieGroups Gon ¢—n ‘commutes. Wehave, inother words, anatural map G+ Aut(G), Theadvantage ofworkingwith¥,isthatitfixestheidentityelemente€G; wwecanthereforeextractsomeofitsstructurebylookingatitsdifferentialat ewe set Ad(@) =(d'¥,).: TG—T.G. (8.2) This isarepresentation ‘Ad:G+Aut(T,G) 63) ofthegroup Gonitsown tangent space, called theadjoint representation of thegroup. This givesathirdcharacterization’: ahomomorphismt prespectsthe adjoint action ofagroupGonitstangentspaceT.Gattheidentity.Inother words, foranyg€Gtheactions ofAd(g)on7,GandAd(p(g))onT.Hmust commute with thedifferential (dp),: 7,G~»T,H, ie.thediagram 7,6TH TGay Tell ‘commutes; equivalently, foranytangent vector v€7,G, Ap(AdKg\(o)) =AdLo(a))(dp(0). 84) This isnice, butdoes notyetanswer ourquestion, forpreservation ofthe adjoint representation Ad:G+Aut(7,G)still involves themap ponthegroup,Gitself,andsoisnotpurelyacondition onthedifferential (dp),.Wehaveinstead t0goonestepfurther, andtake thedifferential ofthemapAd,The group Aut(T.G) being justanopen subset ofthevector space ofendomor- phisms of7,G, itstangent space attheidentity isnaturally identified withEnd(7,G); takingthedifferential ofthemapAdwearriveatamap ad:T,G+End(7,G). 85) ‘Thisisessentially atrilineargadgetonthetangentspace7,G;thatis,wecanview theimage ad(X)(Y) ofatangent vector Yunder themap ad(X)asa "characterization" isnottheright word hee(orinthe preceding cas) since wedonot mean anequivalent condition, bat rather something implied bythecondition that pbe& homomorphism $8.1. LieAlgebras: Motivation andDefinition 107 functionofthetwovariablesXand¥,sothatwegetabilinearmap TG xT.G>T.G. Weusethenotation [,]forthisbilinear map; that is,fora pair oftangent vectors Xand ¥toGat e,wewrite (x,¥ad(xy(¥). (8.6) AAsdesired, themap adinvolves only thetangent space tothegroup Gat ¢,and50gives usourfinal characterization: thedifferential (dp), ofahomo- ‘morphism ponaLiegroup Grespects theadjoint action ofthetangent space toGonitself. Explicitly, thefactthatpanddp,respect theadjoint represen- tation implies inturn that thediagram 7.6i TH 16ay Tel commutes; ie,forany pair oftangent vectors Xand ¥toGat ¢, Apa XV(Y) =adldp (X)}dp). co) or,equivalently, pA, YD)=dp(X) dp(YN (88) Althismay befairly confusing (iitisnot,youprobably donotneed to tereading thisbook). Two things, however, should beborne inmind, They are: ()Ieisnot$0bad, inthesense that wecanmake thebracket operation, asdefined above, reasonably explicit. Wedothisirstforthegeneral linear group G=GL,R. Note that inthiscase conjugation extends totheambientlinearspaceE=End(R)=M,RofGL,Rbythesameformula:Ad(g)(M) =aMg"", andthisambient space isidentified with thetangent space 7.G;differentiation inEisusualdifferentiation ofmatrices. Foranypair oftangent vectors Xand ¥toGL,Rate,let:-+Gbeanarewith7(0)=eandtangent vector ¥/(0) =X.Then ourdefinition of[X, ¥]ithat a [YY]=ad(an(y)=il(adoro Applying theproduct ruletoAd(()(¥) =(0¥7(9) this is =1 YY) +y10)- ¥-(—y10)"* -y'O)-7(0"") aX Yex, Which,ofcourse,explainsthebracketnotation.Ingeneral,anytimeaLie {roupispivenasasubgroupofagenerallineargroupGLP,wecanviewits 108, 8.LieAlgebrasandLieGroups tangent space T,Gatthe identity asasubspace ofthe space ofendomorphisins ofRP;andsince bracket ispreserved by(diflerentials ofmaps ofLiegroups, thebracket operation on7,Gwillcoincide with thecommutator. (ii)Even ifiwere that bad, itwouldbeworthit,Tisisbecauseitturnsout that thebracket operation isexactly theauswer tothequestion weraised before. Precisely, later inthis lecture wewillprove the Second Principle: LetGandHbeLiegroups, with Gconnected andsimply connected. Alinear map 7,G— T,H isthedifferential ofahomomorphism:G-+Iitandonlyifitpreservesthebracketoperation, inthesenseof(8.8)above. Wearenowalmostdone:mapsbetweenLiegroupsareclassifiedbymapsbetweenvectorspacespreserving thestructureofabilinearmapfrointhevectorspacetoitself.Wehaveonlyonemorequestion(oanswer:whendoes.1vectorspacewiththisadditional structureactuallyariseasthetangentspaceattheidentity toaLiegroup, with theadjoint orbracket product? Happily, wehave theanswer tothisaswell. First, though itisfarfrom clear from our initial definition, itfollows from ourdescription ofthebracket asacommu tator that thebracket isskew-symmetric, ie,[X,Y] =—LY, X].Second, it likewise follows from thedescription of[X,¥]asacommutator that it satisfies theJacobi identity: foranythree tangent vectors X,Y,andZ, (40%20)+0%(2,XI]+(2,0%,9)=0. We thus make the Definition 8.9.4Liealgehragisavectorspacetogetherwithaskew-symmetric.bilinear map CL. kaxa-9 satisfying theJacobi identity Weshould take amoment outhere tomake oneimportant point. Why, youmight ask,dowedefine thebracket operation interms oftherelatively difficult operations Adand ad,instead ofjust defining [X,Y]tobethe cominutator X-¥ —Y-X? The answer isthat the“composition” X-¥of elementsofaLiealgebraisnotwelldefined.Specifically, anytimeweembed 1Liegroup Ginageneral linear group GL(V), wegetacorrespondingembedding ofitsLiealgebra ginthespace End(V), andcantalk about the composition XY €End(V) ofelements ofqinthis context; butitmust be borne inmind that thiscomposition X-¥willdepend ontheembedding of {9,and forthat matter need noteven beanelement ofg.Onlythecommutator X-¥—¥-Xisalwaysan elementofindependent ofthe representation,The terminologysometimesheightenstheconfusion:forexample,whenwespeak ‘ofembeddingaLiealgebrainthealgebraEnd(V)ofendomorphisms ofF,the wordalgebra may mean twovery different things. Ingeneral, when wewant 81, LieAlgebras: Motivation and Definition 109 torefer totheendomorphisms ofavectorspaceV(resp.R°)asaLiealgebra, ‘wewill write gi(V) (resp. gl,R) instead ofEnd(V) (resp. M,R). Toreturn toourdiscussion ofLiealgebras, amap ofLiealgebras isalinear‘mapofvectorspacespreserving thebracket,inthesenseof(8.8);notionslikeLiesubalgebra aredefined accordingly. Wenote inpassing onething that will turn outtobesignificant: thedefinition ofLiealgebra does notspecify the field. Thus, wehave real Liealgebras, complex Liealgebras, etc, alldefined inthesame way, andinaddition, given arealLiealgebra wemay associatetoitacomplexLiealgebra,whoseunderlying vectorspaceisq@Candwhosebracket operation isjust thebracket ongextended bylineacity Exercise8.10*.Theskew-commutativity andJacobiidentityalsofollowfromthenaturality ofthebracket (8.8), without using anembedding ingl(V) (a)Deduce theskew-commutativity [X, X]=Ofrom that factthat anyXcan‘bewrittentheimageofavectorbydp,forsomehomomorphism p:R-»G.(Gee §8.3 fortheexistence ofp.)(b)Giventhatthebracketisskew-commutative, verifythattheJacobiidentity,isequivalent totheassertion that ad=d(Ad),: 9-+End) preserves thebracket. Inparticular, adisamapofLiealgebras. Tosum upourprogress sofar:taking forthemoment onfaith thestate- iments made, wehave seen that (@thetangent spacegattheidentitytoaLiegroupGisnaturallyendowed with the structure ofaLiealgebra; (i)ifGand HareLiegroups with Gconnected and simply connected, themaps from GtoHareinone-to-one correspondence with maps of theassociated Liealgebras, byassociating top:G-» Hits differential ead. Ofcourse, wemake the Definition 8.11. Arepresentation ofaLiealgebra gonavector space Vis simply amap ofLiealerbras Bi:9-+(V) =End(V), ie,alinear map that preserves brackets, ofanaction ofqonVsuch that LX, YI}=XV)—FX). Statement i)aboveimpliesinparticular thatrepresentations ofaconnected ‘andsimply connected Liegroup areinone-to-one correspondence with repre- no 8.LieAlgebras and LieGroups sentations ofitsLiealgebra, This is,then, thefirststep oftheseries of reductions outlined intheintroduction toPart II. Atthispoint, afewwords areinorder about therelation between repre- sentations ofaLiegroupandthecorresponding representations ofitsLie algebra.Thefistremarktomakesabouttensors.RecallthatfVandWare representations ofaLiegroup G,thenwedefine therepresentation V@Wto bethevector space V@Wwith theaction ofGdescribed by ae Ow)=O alw). “Thedefinition forrepresentations ofaLiealgebra,however,isquitedifferent. Forone thing, ifgistheLiealgebra ofG,sothat therepresentation ofGon thevector spaces Vand Winduces representations ofgonthese spaces, we wantthetensorproduct oftherepresentations VandWofgtobethe representation induced bytheaction [email protected] that {y,}isanarcinGwith 7=eand tangent vector yo=Xeg.Then bydefinitiontheactionofXonVisgivenby a Xo)=4.yo) and similarly forw€W;itfollows that theaction ofXonthetensorproduct v@wis xee@~=4) gen=F], moen d d~(i{_news oo(4]vm), X0Ow)=X]Ow+ OX 12) This, then, ishow wedefine theaction ofaLiealgebra gonthetensor product oftworepresentations ofThis describes aswellother tensor: frexample, ifVisarepresentation ofthegroup G,v€Visanyvector andv?€Sym? V itssquare, then forany9€G, a0") =ato)”. Ontheotherhand,ifVisarepresentation oftheLiealgebragandXegis any element, wehave XW) =2-0-X(0) 619 (One further example: ifp:G-»GL(V) isarepresentation ofthegroupG,the dual representation p': G+GL(V*) isdefined bysetting #(Q)= "0191. VO V8 Ditferentiating this, wefind that ifp:4-+glV) isa representation of@Lie $82.ExamplesofLieAlgebras ut algebra g,thedual representation ofgonV*willbegivenby BX) ="X)="p(XV8VE. 14) Asecond and related point tobemade concerns terminology. Obviously, ‘when wespeak oftheaction ofagroup Gon avector space Vpreserving someextrastructureon¥,wemeanthatliterally:forexample, ifwehaveaquadratic.formQon¥,tosaythatGpreservesQmeansjustthat (910), o()) =Q(0,w), YoeGand o,we V. Equivalently, wemean that theassociated action ofGonthevector space Sym?¥* fixes theelement Q.€Sym?V*. Butbytheabove calculation, theactionoftheassociated LiealgebragonVsatisies Qo, Xo) +O(X(0}, =O, WKegandrwev (8.15) or,equivalently, Q(e, X(c)) =0forallXe and v€V;inother words, the induced action onSym?V* killstheelement Q.Bywayofterminology, then, wewillngeneral saythat theaction ofaLiealgebraonavectorspacepreserves somestructurewhenacorresponding Liegroupactiondoes.Thenext section willbespent ingiving examples. In§8.3wewillestablish thebasic relations between Liegroups and their Liealgebras, (othe point where wecan prove theFirst and Second Principles above. The furtherstatementthatanyLiealgebraistheLiealgebraofsomeLiegroupwillfollowfrom thestatement (seeAppendix E)thatevery Liealgebra may beembedded inglR. Exercise 8.16*, Show thatifGisconnected theimage ofAd:G+GL(q) isthe adjoint form ofthegroup Gwhen that exists. Exercise 8.17%. Let Vbearepresentation ofaconnected Liegroup Gand :g.-+ End(V) thecorresponding map ofLiealgebras. Show that asubspace WofVisinvariant byGifand only iitiscarried into itself under theactionoftheLiealgebrag,i,p(X)(W)&WforallXing,Hence,VisirreducibleoverGifandonlyifitisirreducible overg §8.2. Examples ofLieAlgebras Westart with theLiealgebras associated toeach ofthegroups mentioned in Lecture 7.Each ofthese groups isgiven asasubgroup ofGL(V) =GL,R, so their Liealgebras willbesubspaces ofEnd(V) =gl,R. ‘Consider frstthespecial linear group SL,R. If{4,)isanarcinSL,R with d= andtangent vector Ap=Xatt=0,then bydefinition wehave for any basis ¢1,...,€,0fV=RY, Ales) A Alen) eyAA ey ry 8LieAlgebrasandLieGroups ‘Taking thederivative and evaluating at«~0wehave bytheproduct rule ao-f]tatennowAled) HDer ar axlanrne, =Trace(X)-(e, Ao Aey) ‘The tangent vectors toSL,Rare thus allendomorphismsoftrace0;comparing dimensions wecanseethat theLiealgebra sR isexactly thevector spaceof tracelessnxmmatrices. ‘Theorthogonal andsymplectic cases aresomewhat simpler. Forexample, theorthogonal group O,R isdefined tobetheautomorphisms Aofan n-dimensional vector space Vpreserving aquadratic form Q,sothatif{4,) isan arcinQ,R with Ay=Jand A,=Xwehave orevery pair ofvectors 0, wer QALo),Adw))=Qo,w). Taking derivatives, weseethat CX(0),»)+Oo,X(w))=0 818) forall v,w€V;this isexactly thecondition that describes theorthogonal Liealgebraso,R=0,R.Incoordinates ifthequadraticformQisgivenonV=RY as Ql0, w)="0Mow (8.19) forsome symmetric nxnmatrix M,then aswehave seen thecondition on AeGL,R tobeinO,Ris that ‘AM-A=M, (820) Differentiating, thecondition onanmxnmattin XtobeintheLiealgebra‘0,8oftheorthogonal groupisthat OMEMEX90. 21NotethatifMistheidentitymatrix—ie, @isthe“standard” quadraticform Q(o, w)='o-w onR*—then thissays thatsoft isthe subspace ofskew- symmetricnxnmatrices.Toputitintrinsically, intermsoftheidentification ‘ofVwith V*given bythequadratic formQ,andtheconsequentidentification End(V)=V@V*=V@FV,theLiealgebras0,R<End(V)isjustthesub- space MV&V@Vofskew-symmetric tensors: 20,8=AVcEnd(V)=V@V. (822) AAllof theabove, with theexception ofthelastparagraph, works equallywellodescribetheLiealgebrasp,4RoftheLiegroupSpa9Roftransforma-tions preserving askew-symmetric bilinear form Q;that is,sp24R isthe subspace ofendomorphisms of¥satisfying (8.18) forevery pair ofvectors t,we¥,or.iFQisgivenbyaskew-symmetric 2nx2nmatrixMasin(8.19),the $82. Examples ofLieAlgebras ns space ofmatrices satisfying (8.21). The one statement that hastobesubstan- tially modified isthelast one ofthelast paragraph: because Qisskew- symmetric, condition (8.18) isequivalent tosaying that 2X(0),w)=XW),0) forallv,weV5thus, interms oftheidentification ofVwith V*given byQ,theLiealgebra sp,,R¢End(V)=V@V*=V@Visthesubspace sym’¥c V@V: Spa,Ft =Sym*V cEnd(V) =V@V. (823) Exercise 824*. With Qastandard skew form, sayofExercise 7.3,describe SpagR and itsLiealgebra spa,R (assubgroup ofGL,R and subalgebra of ‘lzaR). Do&corresponding calculation forSOR. ‘OnemoresimilarexampleisthatoftheLiealgebrau,oftheunitarygroupU(n); byasimilar calculation wefindthat theLiealgebra ofcomplex linearendomorphisms ofC*preservingaHermitianinnerproductHisjustthespace ‘ofmatrices Xsatisfying H(X(o), w)+Ho, X(w)) =0,Yo,weVs ifHisgiven byH(o, w)='B-w, thisamounts tosaying that Xisconjugate skew-symmetric, ie,that‘¥=—X. Exercise 8.25, Find theLiealgebras ofthe realLiegroups SL,C andSL,H— theelements inGL,H whose realdeterminant ist. Exercise 8.26. Show thattheLiealgebras oftheLiegroups B,andN,intro- duced in§7.2arethealgebra b,Rofupper triangular nxnmatrices andthealgebran,Rofstrictlyuppertriangular nxnmatrices,respectively. Ifisacomplex Liegroup, itsLiealgebra isacomplex Liealgebra. Just asin thereal case, wehave thecomplex Liealgebras gl,C, #4,C, #0,C, and ‘Pax oftheLiegroups GL,C, SL,C, SOC, andSpz,C. Exercise 8.27. LetAbeany (real orcomplex) algebra, notnecessarily finite dimensional, oreven associative. Aderivation isalinear map D: A»Asatis- fying theLeibnitz rule D(ab) =aD(b) +D(a) (@)Show that thederivations Der(4) form aLiealgebra under thebracket [D,E]=DoE—EoD.ItAisfinitedimensional, soisDer(A). {(b)Thegroup ofautomorphisms ofAisaclosed subgroup Gofthegroup GL(A)oflinearautomorphisms ofA.ShowthattheLiealgebraofGis Der(A). (9Ifthealgebra AisaLiealgebra, themap A+Der(A), X+-+Dy, where D,(¥) =[X, ¥].isamap ofLiealgebras. na 8.LieAlgebrasandLieGroups Exercise 828°. IfgisaLiealgebra theLiealgebra automorphisms ofgform‘LiesubgroupAut(g)ofthegenerallineargroupGL(a). (a)Show that theLiealgebra ofAut(q) isDer(a).IfG isasimply connectedLiegroupwithLiealgebrag,themapAut(G)-+Aut(q)byg+-+dois‘one-to-one andonto, giving Aut(G) thestructure ofaLiegroup with Lie algebra Der(g). (b)Show that theautomorphism group ofanyconnected Liegroup isaLie subgroup oftheautomorphism groupofitsLiealgebra Exercise 8.29*. For any manifold M,theC*vector fields onMform aLie algebra o(M),as follows: avector field vcanbeidentified with aderivation of thering AofC*functions onM,with v(f) thefunction whose value ata point xofMisthevalue ofthetangent vector v,onfatx.Show that the vector fields onMform aLiealgebra, infactaLiesubalgebra oftheLie algebra Der(A). IfaLiegroupGactsonM,theG-invariant vectorfieldsform aLiesubalgebra vgMofo(M). Ifthe action istransitive, theinvariant vector fields form afinite-dimensional Liealgebra. IfGisaLiegroup, v9(G)=TGbecomesaLiealgebrabytheabove process.Showthatthisbracketagreeswiththatdefinedusingtheadjointmap (8.6).Thisgivesanotherproofthatthebracketisskew-symmetric andsatisfiesJacob's identity. §8.3. The Exponential Map Theessential ingredient instudying therelationship between aLiegroup G and itsLiealgebra qistheexponential map. This may bedefined invery straightforward fashion, using thenotion ofone-parameter subgroups, whichwestudynext.SupposethatX«gisanyelement,viewedsimplyasatangentvector toGattheidentity. Foranyelement g€G,denote bym,:G-+Gthe ‘map ofmanifolds given bymultiplication onthelftbyg.Then wecandefine1vectorfieldronalofGsimplybysetting x(a) =(m,)4(X)- This vector field isclearly invariant under [efttranslation (ie, itiscarried intoitself under thediffeomorphism m,forall g),anditisnothard toseethat thisgives anidentiticationofqwiththespaceofalleft-invariant vectorfields onG.Under this identification, thebracket operation ontheLiealgebra 9corresponds toLiebracketofvectorfields;indeed,thismaybeadoptedasthedefinition oftheLiealgebra associated toaLiegroup (f.Exercise 8.29) For ‘our present purposes, however,'all weneed toknow isthat oexists and is Jeft-invariant. Given anyvector field »onamanifold Mand apoint peM,abasictheoremfromdifferential equationsallowsustointegratethevectorfield.This §8.3.TheExponential Map us ives adifferentiable map g:1 >M,defined onsome open interval Icon- taining 0,with (0)=p,whosetangentvectoratanypointisthevector assigned (0thatpoint byo,ie,stich that 9 =99) forall¢inI.Themapgisuniquelycharacterized bytheseproperties. Nowsuppose themanifold inquestion isaLiegroup G,thevector field theField1associated toanelementX€g,andptheidentity.Wearrivethenatamap9:1-+G;weclaim that, atleast where @isdefined, itisahomomorphism, ic, (5+1) =@(s)o(0) whenever 5,t,and 5+¢areinI.Toprove this, fixsand Jet¢vary; that is,consider thetwo arcsaandfigivenbya(t)=(9)@(t)and fle)=96+1).Ofcourse,a(0)~(0},andbytheinvariance ofthe vector Feld by,weseethat thetangent vectors satisfy a'() =vy(a(t)) andf') =vx(()forallBytheuniqueness oftheintegralcurveofavectorfieldonamanifold,wwededucethata(()=AteforaltFrom thefactthat @(5+1)=@(3)@(0) forall sand1neat0,itfollowsthat extends uniquely toallofR,defining ahomomorphism oeRG with(0) =o<(0X0) =(Muy) y(X)forallt Exercise 8.30. Establish theproduct ruleforderivatives ofarcs ina LiegroupG:ifwand farearcsinGandy(t)=a(t)AC,then (0)=dragBO)+dp, where foranyg€G,themap m,(resp. n,):G—>Gisgiven byleft(resp. right) multiplication byg.Usethisogiveanotherproofthatgisahomomorphism. Exercise 8.31. Show that yyisuniquely determined bythefactthat itis a homomorphism ofRtoGwith tangent vector g4(0) attheidentity equal toX.Deducethatif¥:G+HisamapofLiegroups,then95,x=Vox. The Liegroup map qy:RG iscalled theone-parameter subgroup ofG with tangent vector Xattheidentity. The construction ofthese one-parameter subgroups foreach Xamounts totheverification oftheSecond Principle of $81forhomomorphisms from RtoG.The fact that there exists such a ‘one-parameter subgroup ofGwith anygiven tangent vector attheidentity is crucial. Forexample, itisnothard tosee(wewilldothisinamoment) that these one-parameter subgroups fillupaneighborhood oftheidentity inG,whichimmediately impliestheFirstPrincipleof§8.1.Tocarrythisout,wedefine theexponential map exaG by exp(X)=ex(1). (8.32) 6 £.LieAlgebras andLieGroups Notethatbytheuniquenessof@y,wehave unl oxi: £0thatthe exponential map restricted tothe lines through theorigin in9 fives theone-paraineter subgroups ofG.Indeed, Exercise 8.31 implies the characterization: Proposition 8.33, The exponential mapis theunique map from 90Gtaking O toewhose diferential attheorigin (x.)oiToa=9+1.6=9 isthe identity, andwhose restrictions tothe lines through theorigin in are one-parameter subgroups ofG. ‘This inparticular implies (cf.Exercise 8.31) that theexponential map is natural, inthesense thatforanymap y:G-» HofLiegroups thediagram 6H commutes Now, since thedierential ofthe exponential map atthe origin in9isan isomorphism, theimage ofexpwilcontain aneighborhood oftheidentity in G.1fG isconnected, thiswillgenerate allofG;from thisfollows theFirst Principle: ifGisconnected, then themap isdeterminedbyitsdifferentialdy), attheidentity Using (8.32), wecanwrite down theexponential map veryexplicitly inthe case ofGL4R, and hence oranysubgroup ofGL,R. Wejust usethestandard power series forthe function e%and set,forX€End(V}, epee xeeey 634) Observe that thisconverges andisinvertible, with inverse exp(—X), Clearly, thedifferential ofthismap from toGattheorigin istheidentity; andbythestandardpowerseriescomputation, therestriction ofthe map toanyline through theorigin ig iaone-pararyper subgroup ofG.Thu, themay coincides with theexponential asdtined orginally, and bynatuality he sani isirue forany subgroup ofG.(Note that, aswehave pointed out, the individual termsintheexpression ontherightof(8.34)areverymuchdepen- dent oftheparticular embedding ofGinageneral linear group GL(V) and correspondingly of qnEnd(V), even though thesum onthe right in(8.34)is not) This explicit form oftheexponential map allows ustogive substance to $8.3. The Exponential Map “7 theassertion that “the group structure ofGisencoded intheLiealgebra.” Explicitly, weclaim thatnotonly dotheexponentialsexp(X)generateG,but forXand Yinasufficiently small neighborhood oftheorigin ing,wecan write down theproduct exp(X)-exp(¥) asanexponential. Todothis, we introduce firstthe“inverse” oftheexponential map: forg€GcGL,R,weset 1 @-y log)=(9=~OOD eg Ofcourse, thiswillbedefined only forgsufficiently close totheidentity inG; ‘butwhere itisdefined itwillbeaninverse totheexponential map. Now, wedefineanewbilinearoperationongl,R:weset X+¥ =loglexp(X)-exp()). Wehavetobecarefulwhatwemeanbythis,ofcourse;Wesubstitute forgin theexpression aboveforlog(g)thequantity exp(x)erry(14+ ve)(1yee ) aro ne(Sere Ben,2 2 being careful, ofcourse, topreserve theorder ofthefactors ineach product.Doingthis,wearriveat xeven srs(FPP yereZ))y- 2 2 2 aX YEG YDS Observe inparticular that theterms ofdegree 2inXand ¥donot in- ‘volvethesquares ofXand¥ortheproduct X'-Yalone,butonlythecom-‘mutator. Infact, this istrue ofeach term intheformula, ic,thequantityloglexp(X)-exp()) canbeexpressedpurelyintermsofX,Y,andthebracket‘operation; theresulting formula iscalledtheCampbell-Hausdorff formula {although theactual formula inclosed form wasgiven byDynkin). Todegree three, itis Xey=X4+V +40 VIE SOGOX VIDS ALOK XI te Exercise 8.35*. Verily (and find thecorrect signs in)thecubic term ofthe Campbell- Hausdorff formula. Exercise8,36.Provetheassertion ofthelastparagraph thatthepowerserieslog(exp(X)- exp(¥)) canbeexpressed purelyintermsofX,¥,andthebracket operation Exercise 8.37. Show that forXand Ysufficiently small, thepower series Joglexp(X)-exp(¥)) converges. 18 8.LieAlgebrasandLieGroups Exercise 8.38*. (a)Show that there isaconstant Csuch that forX,¥€gly Xe =X+V4(X, ¥]+E,where BES CUXE +WYP. (6)Show that exp(X +¥)=limy..(exp(X/n)-exp(Y/n)f. (6)Show that , x Y x ny aunts=m(en(*)-90(")-20(—*)e0(-2)) Exercise 8.39. Show that ifGisasubgroup ofGL,R, theelements ofits Liealgebra arethe“infinitesimal transformations” ofGinthesense ofvon Neumann, ie,they arethematrices ing{R which canberealized aslimits timA=,AveG6,>0,690. Exercise840,ShowthatexpissurjectiveforG=GL,CbutnotforG=GL;R ifn> tor forG=SLC. BytheCampbell-Hausdorff formula, wecan notonly identify alltheclementsofGinaneighborhood oftheidentity,butwecanalsosaywhattheirpairwiseproductsare,thusmakingprecisethesenseinwhichganditsbracket operationdeterminesGanditsgrouplawlocally.Ofcourse,wehavenot written aclosed-form expression fortheCampbell-Hausdorff formula; but,aswewillsceshortly,itsveryexistenceissignificant.(Forsuchaclosedform, see[Sel, 164.8]) ‘Wenow consider another very natural question, namely, when avectorsubspaceh<qistheLiealgebraof(ic,tangentspaceattheidentityto)animmersedsubgroupofG.Obviously, anecessarycondition isthatlisclosedunderthebracketoperation; weclaimherethatthisissufficientaswell: Proposition 8.41, LetGbeaLiegroup, 9itsLiealgebra, andh9aLie subalgebra. Then thesubgroup ofthegroup Ggenerated byexp(b) isan immersed subgroup Hwithtangent space 7,1 =). Proor. Note that thesubgroup generated byexp(b) isthesame asthesub- groupgenerated byexp(U),whereUisanyneighborhood oftheorigininb.Itwillsuffice, then (seeExercise 8.42), toshow thattheimage ofunder the exponential map is“locally” closed under multiplication, ie,thatforasuffi-cientlysmalldiscA<6,theproductexp(A):exp(A) thatis,thesetofpairwiseproducts exp(X)-exp(Y) forX,¥€A)iscontained intheimage of under theexponential map, Wewilldothisunder thehypothesis thatGmay berealized asasubgroup ofagenerallineargroupGL,R,sothatwecanusetheformula(8.34)fortheexponential map. This isaharmless assumption, given thestatement (tobe proved inAppendix E)that any finite-dimensional Liealgebra may be §83. The Exponential Map 119 embedded intheLiealgebra glyR: thesubgroup ofGL,R generated by exp(g) willbeagroup isogenous toG,and, asthereader caneasily check,provingtheproposition foragroupisogenoustoGisequivalent toprovingittor G. ItthussufficestoprovetheassertionincasethegroupGisGL,R.Butthis isexactly thecontent oftheCampbell- Hausdorff formula a When applied toanembedding ofaLiealgebra ginto ly,wesee, in particular, that every finite-dimensional Liealgebra istheLiealgebra ofaLie‘group.Fromwhatwehaveseen,thisLiegroupisuniqueifwerequiteittobesimply connected, and then allothers areobtained bydividing thissimply connected model byadiscrete subgroup ofitscenter. Exercise£.42*,SupposeGyisanopenneighborhood oftheidentityinaLiegroup Gsuch that Go:Gy©Gyand Ga!=Go.Suppose Hpisaclosed sub- manifold ofGysuch thatHo:Hy=Hand Ha!=Hy.Show thatthesubgroup HofGgenerated byHgisanimmersed LiesubgroupofG. ‘Asafairlyeasyconsequence ofthisproposition, wecanfinallygiveaproof oftheSecond Principle stated in§8.1, which wemay restate as, Second Principle. LetGandHbeLiegroupswithGsimplyconnected,andlet sand lybetheir Liealgebras. Alinear mapa:9>isthediferentialofamap A:G—HofLiegroupsifandonlyif«isamapofLiealgebras, Proor.Toseethis,considertheproductGxH.ItsLiealgebraisjust9®b.Let|[email protected] thataisamapofLicalgebrasisequivalent tothestatementthatjisaLiesubalgebra ofg@b; andgiven this, bytheproposition there exists animmersed Liesubgroup, J.¢G xHtwith tangent space T.J=j Look now atthemap :J +Ggiven byprojection onthefirst factor. Byhypothesis, thedifferential ofthis map dx, ~gisanisomorphisin, sothat themap J+Gisanisogeny; butsince Gissimply connected itfollows thatisanisomorphism. Theprojection n:G=J-»HonthesecondfactoristhenaLiegroupmapwhosedifferential attheidentityis« a Exercise 8.43*. If-+9/isahomomorphism ofLiealgebras with kernel show thatthekernel Hofthecorresponding map ofsimply connected Lie groups G+ G’isaclosed subgroup ofGwith Liegroup h.This does not extend tonon-normal subgroups,ie.tothesituationwhenfisnotthekernel ofahomomorphism: give anexample ofanimmersed subgroup ofasimply connected Liegroup Gwhose image inGisnotclosed. Exercise 8.44, Use theideas ofthis lecture toprove theassertion that a compact complex connected Liegroup Gmust beabelian: 120 8LieAlgebrasandLieGroups (a)Verify that themap Ad:G-+Aut(T,G)<End(T,G)isholomorphic, and, therefore (bythemaximum principle), constant, (b)Deduce that if¥,isconjugation byg,then d'¥, istheidentity, so "Y,(exp(X)) =exp(t'F,(X)) =exp(X) forallX€7,6, which implies that Gis abelian. (©)Show thattheexponentialmapfrom7,GtoGissurjective,withthekernel alattice A,soG=T,G/A isacomplex torus. LECTURE 9 Initial Classification ofLieAlgebras Inthislecture wedefine vatious subclasses ofLicalgebras: nilpotent, solvable, semi-simpesete,and provebasicfatsabouttheirrepresentations, Thediscussionsetielyelementary(largelybecausethehardtheoremsarestatedwithoutprooffornow)therearenoprerequisites beyond linear algebra. Apart from giving these bate defitons thepurpose oftheleture itlargely tomotivate thenarrowing ofourfocus to semisimple algebras thatwilltakeplace inthesequel. nparticular, thefrstpartof §9.3islogically themost important forwhat follows. $9.1: Rough classification ofLiealgebras $92: Engels Theorem and Lie's Theorem §9.3; Semisimple Liealgebras 4: Simple Lialgebras 99.1. Rough Classification ofLieAlgebras Wewillgive,inthissection, apreliminary sortofclassification ofLie algebras, reflecting thedegreetowhichagivenLiealgebragfailstobeabelian. Aswehave indicated, thegoal ultimately istonarrow ourfocus onto semisimple Lie algebras. Beforewebegin,twodefinitions, bothcompletly straightforward: First, wedefinethecenterZ(q)ofaLiealgebragtobethesubspaceofgofelements: Xeqsuch that [X, ¥]=0forallYeg. Ofcourse, wesaygisabelian ifall brackets are zero. Exercise9.1.LetGbeaLiegroup,9itsLiealgebra.ShowthatthesubgroupofGgenerated byexponentiating theLiesubalgebra Z(q) istheconnected component oftheidentity inthecenter Z(G) ofG. 1m 9.Initial Classification ofLieAlgebras Next, wesaythataLiesubalgebra h<gofaLiealgebragisanidealifit satisfies the condition [XK Yeh forall Keb Ves. Justascotinected subgroups ofaLiegroupcorrespond tosubalgebras ofitsLiealgebra, thenotion ofideal inaLiealgebra corresponds tothenotion of, normal subgroup, inthefollowing sense: Exercise 9.2.LetGbeaconnected Liegroup, H<Gaconnected subgroup andqandfytheir Liealgebras, Show that HYisanormal subgroup ofGifand.onlyifisanidealofg, Observe alsothatthebracket operation onqinduces abracket onthequotient spacea/bifandonlyifisanidealin9This, inturns, motivates thenext bitofterminology: wesay that aLie algebraqissimpleifdimg>anditcontainsnonontrivialideals.Bythelastexercise,thisisequivalent tosayingthattheadjointformGoftheLiealgebra{has nonontrivial normal Liesubgroups. Now, toaltempt (0classify Liealgebras, weintroduce two descending.chainsofsubalgebras. Thefirstisthelowercentralseriesofsubalgebrasdefined inductively by _%= Loa and 9 =[9%-r9]- Note thatthesubalgebras 2%areinfactideals ing.Theother series iscalled thederived series {g}; itisdefined by a q=[9,0] and Ho=(99, #9). Exercise 9.3.UsetheJacobi identity toshow that 9% isalso anideal ing More generally, it}isanideal inaLiealgebra g,show that (9,}]isalso an ideal ing;hence all91) areideals inq Observe thatwehave 99<Mqforallk,with equality when k=I;we often write simply 9for29=9gandcallthisthecommutator subalgebra ‘We now make the Definitions. (Wesaythatgisnilpotentif%g=0for somek (il)Wesaythat gissolvable it9*q =0forsome k. 59.1.RoughClassification ofLieAlgebras 23 (il)Wesaythatgisperfectif9g=9thisisnotaconceptwewillusemuch)(iv)Wesaythatgissemisimple ifghasnononzero solvable ideals. The standard example ofanilpotent Liealgebra isthealgebra n,R of strictly upper-triangular nxmmatrices; inthiscase thekthsubalgebra 9 inthelower central series willbethesubspace 1,48 ofmatrices A=(a,) suchthat aj;=0whenever j<i+k, ie,that arezero below thediagonalandwithinadistancekofitineachcolumnorrow.(Intermsofacompleteflag{V;}asin§7.2, thesearejusttheendomorphisms thatcarryintoVj-x-1) ItfollowsalsothatanysubalgebraoftheLiealgebran,Rislikewisenilpotent; ‘wewillshow later that anynilpotent Liealgebra isisomorphic tosuch a subalgebra. Wewillalsoseethat ifaLiealgebragisrepresentedonavector space V,such that each element acts asanilpotentendomorphism, thereisa basis forVsuch that, identifying gf(V) with gl,R, 9maps tothesubalgebra nReghR Similarly, astandard example ofasolvable Liealgebra isthespace b,Rof ‘upper-triangular nxnmatrices;inthisLiealgebrathecommutator 6,Ris thealgebra nRandthederived series i,thus, O*,R =toe,qR Again, it follows that any subalgebra ofthealgebra b,R islikewise solvable; and wewillprovelaterthat,conversely, anyrepresentation ofasolvableLiealgebra‘onavector space Vconsists, interms ofasuitable basis, entirely ofupper-triangular matrices(ie,givenasolvableLiesubalgebragofgl(V),thereexists ‘basis forVsuch that under thecorresponding identification ofgl(V) with1,R,thesubalgebra 9iscontained inby<gl,Itisclear from thedefinitions that theproperties ofbeing nilpotent or solvable areinherited bysubalgebras orhomomorphic images. Wewillsce thatthesame istrue forsemisimplicity inthecase ofhomomorphic images, though notforsubalgebras. Note that qissolvable ifandonly ifghasasequence ofLiesubalgebras 8=0>91>> =0,suchthatgrogisanidealimgyandgi/tuesixabelian, Indeed, ifthisisthecase, onesees byinduction thata<4,forall i(One may also refine such asequence toone where each quotient g/d is‘onedimensional) Itfollowsfromthisdescription thatif}isanidealinaLiealgebrag,thengissolvableifandonlyifandq/yaresolvableLiealgebras.(The analogous assertion fornilpotent Liealgebras isfalse: theideal ngis nilpotent intheLiealgebra b,ofupper-triangular matrices, andthequotientisthenilpotentalgebrab,ofdiagonal matrices, butb,isnotnilpotent) Ifgis theLiealgebra ofaconnectedLiegroupG,thengissolvableifandonlyif thereisasequenceofconnected subgroups, each normal inG(orinthenext inthe sequence), such that thequotients areabelian Inparticular, thesum oftwo solvable ideals inaLiealgebra gisagainsolvable[notethat(a+bY&a/(a6)}.Itfollowsthatthesumofallsolv-ableideals ingisamaximal solvable ideal, called theradical ofganddenoted Radi(q). The quotient g/Rad{(q) issemisimple. Any Liealgebra 9thus fitsinto anexact sequence 14 9,Initial Classification ofLieAlgebras 0Rad(g)-»9+g/Rad(g)+0 04) where thefirst algebra issolvable and thelast issemisimple, With this somewhat shaky justification (but seeProposition 9.17), wemay saythat to study therepresentation theory ofanarbitrary Liealgebra, wehave to understand individually therepresentation theories ofsolvable and semi- simple Liealgebras. Ofthese, theformer isrelatively easy, atleast asregards irreducible representations. Thebasic factabout them—that anyirreduciblerepresentation ofasolvableLiealgebraisonedimensional—will beprovedlater inthislecture, Therepresentation theory ofsemisimple Liealgebras, on theother hand, isextraordinarily rich, and itis thissubject that will occupy usformost oftheremainder ofthebook. ‘Another easy consequence ofthedefinitions isthefactthat aLiealgebra issemisimple ifandonly ifithasnononzero abelian ideals. Indeed, thelast‘nonzeroterminthederivedsequenceofideals9*Rad(g)wouldbeanabelianideal ing(cf.Exercise 9.3). Asemisimple Liealgebra canhave nocenter, so theadjoint representation ofasemisimple Liealgebra isfaithful Itisafactthat thesequence (9.4) splits, inthesense that there aresub-algebrasofgthatmapisomorphically ontoq/Rad(a).Theexistenceofsuch«Levidecomposition ispartofthegeneraltheorywearepostponing, ToshowthatanarbitraryLiealgebrahasfaithfulrepresentation (Ado’stheorem),onestarts with afaithful representation ofthecenter, andthen builds arepresen- {ation ofthe radical stepbystep,insertingastringofidealsbetweenthecenter and theradical. Then one uses aspitting togetfrom afaithful representation contheradical tosome representation onalof; thesumofthis representationandtheadjointrepresentation isthenafaithfulrepresentation. SeeAppendixEfordetails.‘Onereason fortheterminology simple/semisimple willbecome clear later inthislecture, when weshow that asemisimple Liealgebra isadirect sum of simple ones, Exercise 9.Every semisimple Liealgebraisperfect.ShowthattheLiegroup ‘ofEuclidean motions ofRhasaLiealgebra gwhich isperfect, ie,99=9, ‘but gisnot semisimple, More generally, if)issemisimple, and Visan irreducible representation ofb,thetwisted product 9=(0,Xlve HX eb} with [(0,X),(w, YY}=(Xw— Yo,[X,YD) isaLiealgebra with 9=9,Rad(q) =Vabelian, andq/Rad(q) =b. Exercise 9.6,(a)Show thatthefollowing areequivalent foraLiealgebra 9:() 9:isnilpotent, (i)There isachain ofideals 9=go>9) >-*->9,=0with 1/4144 contained inthecenter ofg/ays1- (i)There isaninteger nsuch that dX)0ad(Xy)0+0ad(NQ)(Y)=[Kay(Xa004XeVY)=0 forallX15...) XqpYing, $9.2. Engels Theorem and Lie's Theorem as (b)Conclude that aconnected Liegroup Gisnilpotent ifand only ifitcanberealizedasasuccession ofcentralextensions ofabelian Liegroups. Exercise 9.7. IfGisconnected and nilpotent, show that theexponential mapexp:q+Gissurjective, makinggtheuniversalcoveringspaceofG. Exercise 98,Show that thefollowing areequivalent foraLiealgebra 9:()9 issolvable. (ii)There isachain ofideals g=go>9)>" >94=0withs/s,abelian.(ii)Thereisachainofsubalgebras 9=M9>ay>">=0 such thatgy,isanideal ing,,andgy/qis1 isabetian, §9.2. Engel’s Theorem and Lie’s Theorem Wewillnow prove thestatement made above about representations ofsolv- able Liealgebras always being upper triangular. This may give thereader an idea ofhow thegeneral theory proceeds, before wegoback totheconcrete ‘examples thatareourmain concern. The starting point is ‘Theorem 9:9(Enge!'s Theorem). Letg<gl(V) beanyLiesubalgebra such that every X€gia nilpotent endomorphism ofV.Then there exists anonzero vector v6VsuchthatX(o)=0forallX©9, ‘NotethisimpliesthatthereexistsabasisforVintermsofwhichthematrix representativeofeachX¢qisstrictlyuppertriangular:since¢kills0,itwill actonthequotientVofVbythespan ofv,andbyinduction wecanfinda basis 7... forVinterms ofwhich thisaction isstrictly upper triangular.Lifting6,toanyv,€Vandsettingv,=»thengivesabasisforVasdesired. Proor oFTsonem 9.9. One observation before westart isthat ifX€al(V) isanynilpotent element, then theadjoint action ad(X): gl(V)-» gl(V) isnil- potent. This isstraightforward: tosaythat Xisnilpotent istosaythatthereexistsaflagofsubspaces 0.Vy©VyC7©KCKay=VsuchthatX(V)) €Vi-s3 wecan then check that forany endomorphism ¥of¥the‘endomorphism ad(X"(Y) carties¥;intoVisa‘Wenowproceedbyinduction onthedimension ofg,Thefirststepistoshow that, under thehypotheses oftheproblem, 9contains anideal 1of codimension one. Anfact, letI} beanymaximal proper subalgebra; we claim that Ihascodimension one and isanideal. Toseethis, welook atthe adjoint representation ofg;since lisa subalgebra theadjoint action ad(b) of,‘honqpreservesthesubspace <qandsoactsong/.Moreover, byour“observation above, foranyX€had(X) acts nilpotently ongl(V}), hence on9, henceong/h.Thus,byinduction, thereexistsanonzeroelement¥«g/bkilledbyad(X)forallX¢byequivalently, thereexistsanelement¥€gnotinhsuch 126 9,InitialClasiication ofLieAlgebras that ad(X)(¥)€bforallXeb.Butthisistosaythatthesubspacebyofq spannedbybandYis@Liesubalgebra ofg,inwhich6sitsasanidealofcodimension one;bythemaximality ofwehaveb=gandwearedone.Wereturn now totherepresentationofgon¥.Wemayapplytheinduction hypothesis tothesubalgebra ofqfound inthepreceding paragraph to conclude that there exists anonzero vector v¢Vsuch that X(c) =0forall X€by,etWcVbethesubspaceofall such vectors ve.Let ¥beany clement ofgnotin§;since hand ¥span g,itwill suffice toshow that there ‘existsa(nonzero)vectorv€Wsuchthat¥(o)=0.NowforanyveetorweW and any X€b,wehave X(Y())=YX(0)+EX,YT).‘Thefirsttermontherightiszerobecausebyhypothesis weW,X€andsoX(w) =0;likewise, thesecond term iszero because [X,¥}=ad(X)(Y)eb,Thus,X(¥(w))=OforallX€b;wededucethatY(w)€W.Butthismeansthat theactionof¥onVcarriesthesubspaceH¥intoitself;sinceYacts nilpotently ‘on¥,itfollows that there exists avector veWsuch that ¥(o) =0. oO Exercise 9.10*. Show that aLiealgebra gisnilpotent ifandonly ifad(X)is anilpotentendomorphism ofqforeveryX€9. Engels theorem, inturn, allows ustoprove thebasic statement made above that every representation ofasolvableLiegroupcanbeputinupper- triangular form, This isimplied by ‘Theorem 9.11(Lie’s Theorem). Letg<gl(V)beacomplexsolvableLiealgebra. Thenthereexistsanonzerovectorv¢Vthatisaneigenvector ofXforallX€g Exercise9.12.ShowthatthisimpliestheexistenceofabasisforVintermsofwhichthematrixrepresentative ofenchXegisuppertriangular. Proor oFTHroreM 9.11. Once more, thefirststep intheargument istoassertthatgcontainsanidealbofcodimension one.Thistime,sincegissolvableweknowthat2g#g,sothat thequotient a=g/g isanonzero abelian Lie algebra; theinverse image ingofanycodimension onesubspace ofawill then beacodimension one ideal in. Still following thelines oftheprevious argument, wemay byinductionassumethatthereisavectorvp€VthatisaneigenvectorforallX€b,Denote theeigenvalue ofXcorresponding tov9byA(X). Wethen consider thesubspaceWcVofallvectorssatisfyingthesamerelation,ic.,weset W=(ve V:X(o)=A(X)-wYXeb).Let¥nowbeanyelementofgnotinb,Asbefore,itwillsulficetoshowthat Ycarriessomevectorv€Wintoamultipleofitself,andforthisitisenough $9.2. Engels Theorem and Lie's Theorem 127 toshowthat¥carriesWintoitself.Weprovethisinageneralcontextinthefollowing lemma, Lemma9.13.LetbbeanidealinaLiealgebrag,LetVbearepresentation of.and 2:t)-+€ alinear function, Set W=(veV:X(0) =A(X)-0 YXeh}, Then¥(W)<Wforall¥€q. Proor. Letwbeany nonzero element ofW;totestwhether ¥(w) ©WweletXbeanyelementofhandwrite X(YEW) =OCW) +OX, YH) =AX): Y(w)+AEX,YD 14) since[X,¥]eb,Thisdiffersromourpreviouscalculation inthatthesecond term ontheright isnotimmediately seen tobezero; indeed, ¥(w) wil lieinWifandonlyifACX,¥})=0forallXeb.Toverify this, weintroduce another subspace of¥,namely, thespan Uof theimages w,¥(w), ¥2(w),... ofwunder successive applications of¥.This subspace isclearly preserved byY;weclaim that any X€carries Uinto itself aswell. Itiscertainly thecase that bcarries winto amultiple ofitself,andhenceintoU,and(9.14)saysthatjycarties¥(w)intoalinearcombination ‘of¥(w)andw,andsointoU.Ingeneral, wecanseethat§carries ¥*(w) into Ubyinduction: foranyXehwewrite XCPH OW)=FORO) +OX,YT). 9.15) Since X(Y#~"(w))€UbyinductionthefirsttermontherightisinU,and since [X, ¥]ebthesecond term isinUaswell Infact, weseesomething more from (9.14) and(9.15): itfollows that, in terms ofthebasis w,Y(w), ¥2(w),... forU,theaction ofany X€bisuppertriangular, withdiagonalentriesallequalto2(X).tnparticular, foranyX€thetrace oftherestriction ofXtoUisjust thedimension ofUtimes 4(X).Ontheotherhand,foranyelementXei)thecommutator [X,YJactsonU,andbeing thecommutatoroftwoendomorphisms ofUthetraceofthisaction iszero,ItfollowsthenthatA([X,YJ)=0,andwearedone. a Exercise 9.16. Show that any irreducible representation ofasolvable Lie algebra gisone dimensional, and gacts trivially Atleast forirreducible representations, Lie's theorem implies they will allbeknownforanarbitrary Liealgebrawhentheyareknownforthesemisimple‘ase, Infact, wehave: Proposition 9.17. Let qbeacomplex Liealgebra, Qu=e/Rad(q). Every irre-Auciblerepresentation ofqsoftheformV=Vp@L,whereVpisanirreducible 8 9,ital Clasication ofLieAlgebras representation of94,[ie., arepresentation ofgthat istrivial onRad(q)), and Lisa one-dimensional representation PRroor. ByLie’s theorem there isa2€(Rad(g))* such that W={veV:X(0) =A(X)-0 WX©Rad(g)} isnotzero. Apply thepreceding lemma, with h=Rad(g). Since Visirreduc-ible,wemusthaveW=V.Nowextend4inanyway(oalinearfunction on ‘,andletLbetheone-dimensional representation ofgdetermined by4;in otherwords,¥(z)=A(Y)-zforallYeqandzeL.ThenV@L*isarepre-sentation thatistrivialonRad(@),80itcomesfromarepresentation ofax ‘asrequired, o Exercise9.18,Showthatifisasubalgebra ofgthatmapsisomorphically‘onto 9/Radiq), then any irreducible representation ofgrestricts toanirre-duciblerepresentation ofq’,andanyirreducible representation of9’extendstoarepresentation of9. §9.3. Semisimple LieAlgebras Asisclear from theabove, many oftheaspects oftherepresentation theory ‘offinite groups thatwere essential toourapproach arenolonger valid inthe context ofgeneral Liealgebras and Liegroups. Most obvious ofthese is complete reducibility, which wehave seen failsforLiegroups; another isthe factthat notonly cantheaction ofelements ofaLiegroup oralgebra onavectorspacebenondiagonalizable, theactionofsomeelementofaLiealgebramay bediagonalizable under onerepresentation andnotunder another. ‘Thatisthebadnews.Thegoodnewsisthat,ifwejustrestrictourselves to.semisimple Liealgebras, everything isonce more aswellbehaved aspossible, Foronething, wehave complete reducibility again: ‘Theorem 9.19(Complete Reducibility). LetVbearepresentation ofthesemi- simple Liealgebra gand W<Vasubspace invariant under theaction of. ‘Then there existsasubspaceW"<Vcomplementary coWandinvariantunderg, ‘The proof ofthisbasic result willbedeferred toAppendix C. ‘Theotherquestion, thediagonalizability ofelements ofaLiealgebraunder arepresentation, requires alittle more discussion. Recall first thestatement ofJordan decomposition: anyendomorphism XofacomplexvectorspaceVcanbeuniquely written intheform X=X+%, $93.SemisimpleLieAlgebras 129 where X,isdiagonalizable, X,isnilpotent, andthe(wocommute. Moreover, X,andX,may beexpressed aspolynomials inX. Now, suppose that isanarbitrary Liealgebra, X©anyclement, and 1:9-ala€ anyrepresentation. Wehave seen that theimage p(X) need not bbediagonalizable; wemay stillaskhow p(X) behaves with respect totheJordandecomposition. Theanswersthal,ingeneral,absolutely nothingneedbetrue. Forexample, justtakingg=C,weseethatundertherepresentation reo every clement isdiagonalizable, ie,p(X), =p(X), under therepresentation can? ®PENG 9, every element isnilpotent [ie, p(X), =OJ;whereas under therepresentation o((‘) nO) notonly aretheimages p(X) neither diagonalizable nornilpotent, thedia- gonalizable andnilpotentpartsofp(X)arenotevenintheimagep(q)oftherepresentation. Iweassume theLiealgebra 4issemisimple, however, thesituation is radically different. Specifically, wehave ‘Theorem9.20(Preservation ofJordanDecomposition}. Letgbeasemisinple Liealgebra. Foranyelement X€g,there exist X,andX,€ gsuch thatforany representation p:q gl(V) wehave AUX), =0(%) and p(X), =p(X) Inotherwords,ifwethinkofpasinjectiveandgaccordingly asaLiesubalgebra ofql(V},thediagonalizable andnilpotentpartsofany'element Xofre again inqandareindependent oftheparticular representation p, ‘The proofs wewilgive ofthe lasttwo theorems both involve introducing‘objectsthatarenotessentialfortherestofthisbook,andwethereforerelegatethem toAppendix C.Itisworth remarking, however, that another approach‘wasusedclassicallybyHermannWeylthisisthefamousunitarytrick,whichvewilldescribe briefly. ADigression on“The Unitary Trick” Basically, theidea isthat thestatements above (complete reducbility, pre- servation ofJordan decomposition) can beproved readily fortherepresen- tations ofacompact Liegroup. Toprove complete reducbilty forexample, 130 9.InitialClassification ofLieAlgebras ‘wecanproceedmoreorlessjustasinthecaseofafinitegroup:ifthecompactgroupGactsonavectorspace,weseethatthereisaHermitian metriconVinvariant under theaction ofGbytaking anarbitrary metric onVand averaging itsimages under theactionofG.IfGfixesasubspaceWc¥,itwill then fixaswellitsorthogonal complement W!with respect tothismetric (Alternatively, wecan choose anarbitrary complement WV’toW,notneces-sarilyfixedbyG,andaverageoverGtheprojection maptog(W)withkernelW;thisaverage willhave image invariant under G.) How does this help usanalyze therepresentation ofasemisimple Lie algebra? The keyfact here (tobeproved inLecture 26)isthat if9isany complex semisimple Liealgebra, there existsa(unique)realLiealgebragywith complexification gg@C=a,such that thesimply connected form oftheLiealgebragois«compactLiegroupG.Thu,restricting agivenrepresentation‘ofqtoqo,wecanexponentiate toobtain arepresentation ofG,forwhich complete reducibility holds; and wecandeduce from thisthecomplete re- ducibility oftheoriginal representation. Forexample, while itiscertainly not true that anyrepresentation oftheLiegroup SL,R onavector space V admits aninvariant Hermitian metcc (infact, itcannot, unless itis thetrivial representation), wecan {iletp’bethecorresponding (complex) representation oftheLiealgebra StF(i)bylinearityextendtherepresentation p’ofs,Rto.arepresentation p"ofsl;(ii)restricttoarepresentation p”ofthesubalgebra sus,<s1,C;(iv)exponentiate toobtain arepresentation p™oftheUnitary group SU,. Wecannow argue that Ifasubspace W<Visinvariant under theaction ofSLR, itmustbeinvariantunders1,R;andsincesI,C=s1,8@C,itfollowsthat itwillbeinvariantundersl,€;s0ofcourse itwillbeinvariant under sty; and hence itwillbeinvariant under SU,, Now, since SU,iscompact, there willexist acomplementary subspace ”preservedbySU,;wearguethat W’willthen beinvariant under su,; and since s1,C =su,@C,itfollows that itwillbeinvariant under si,€. Restricting, weseethat itwillbeinvariant under s1,R, andexponentiating, itwillbeinvariant under SLR. Similarly, ifonewants toknow that thediagonal elements ofSL,R act semisimply inanyrepresentation, orequivalently that thediagonal elements ofs{,R actsemisimply, onegoes through thesame reasoning, coming down tothe factthatthe groupofdiagonalelementsinsu,isabelianandcompact. $94, Simple LieAlgebras I Ingeneral, most ofthetheorems about thefinite-dimensional represen-{ationofsemisimple Liealgebrasadmitproofsalongtwodiflerentlines:eitheralgebraically, usingjustthestructureoftheLiealgebra;orbytheunitarytrick, thats,byassociatingtoarepresentation ofsuchaLiealgebra.representation ‘ofacompactLiegroupandworkingwiththat.Whichispreferable dependsvery much ontaste andcontext; inthisbook wewillforthemost partgowith thealgebraic proofs, though inthecase oftheWeyl character formula inPart IVtheproof viacompact groups issomuch more appealing ithas tobe mentioned, ‘Thefollowingexercisesincludeafewapplications ofthesetwotheorems. Exercise9.21°,ShowthataLiealgebraqissemisimple ifandonlyifeveryFinite-dimensional representation issemisimple, ie.,every invariant subspace hasacomplement. Exercise 9.22 UseWeyl's unitary trick toshow that, forn>2,allrepresen- tations ofSO,€ aresemisimple, sothat, inparticular, theLiealgebras #0,€ aresemisimple, Dothesame forSp,,€ and sp,,€, n>I,Where does the argument break down forSO,€7 Exercise 9.23. Show that areal Liealgebra qissolvable ifandonly ifthe complex Licalgebra 9@qCissolvable,Similarlyfornilpotentandsemisimple. Exercise9.24*,If}isanidealin@Liealgebra9,showthatgissemisimple ifandonly if}andg/haresemisimple, Deduce thatevery semisimple Liealgebraisadirectsumofsimple Liealgebras. Exercise 9.25%. ALiealgebra iscalled reductive ifits radical isequal toits center. ALiegroup isreductive ifitsLiealgebra isreductive. Forexample,GL,Cisreductive. ShowthatthefollowingaretrueforareductiveLiealgebra 1()9gissemisimple; Gi)theadjointrepresentation ofqissemisimple; (i)9isaproduct ofasemisimple and anabelian Liealgebra; (iv)ghasafinite- dimensional faithful semisimple representation, Infact, each ofthese condi- {ions isequivalent togbeing reductive. §9.4. Simple LieAlgebras There isonemore basic factabout Liealgebras tobestated here; though itsProofwillhavetobeconsiderably deferred,itinformsourwholeapproachtothesubject. This isthecomplete classification ofsimple Liealgebras: ‘Theorem 9.26. With five exceptions, every simple complex Liealgebra isiso-morphictoeithersl,€,£0,€,oFspz4€forsomen. i 9,Initial Clasifiation ofLieAlgebras ‘The fiveexceptions canallbeexplicitly described, though none ispar- ticularly simple except inname; they aredenoted qa,fas€,€7,and ey.We willgive aconstruction ofeach later inthebook (§22.3. The algebras s1,C (forn>1),80,€ (for n>2),and spz,€ arecommonly called theclassical Lie ‘algebras and thecorresponding groups theclassical Liegroups) theother five algebras arecalled, naturally enough, theexceptional Liealgebras.‘Thenatureofthe classification theorem forsimple Liealgebras creates adilemmaastohowweapproach thesubject:manyofthetheoremsaboutsimpleLiealgebrascanbeprovedeitherintheabstract,orbyverifyingtheminturnforeachoftheparticularalgebraslistedintheclassification theorem.Another alternative istodectare that weareconcerned with understanding only therepresentations oftheclassical algebras sI,C, s0,C, andspayC, and verily any relevant theorems just inthese cases. Ofthese three approaches, thelastisin many ways theleat satisfactory; iti, however, theone that weshall forthemost patt take. Specifically, what wewilldo,starting inLecture 11. thefollowing: ()AnalyzeinLectures11-13acoupleofexamples, namely, sls€ andsf,€, ‘onwhat may appear tobeanadhocbasis. {ii)Onthebasis ofthese examples, propose inLecture 14a general paradigin forthestudy ofrepresentations ofasimple(orsemisimple)Liealgebra. (ii) Proceed inLectures 15-20 tocarry outthisanalysis fortheclassical algebras s1,C, s0,C, and sp2,C. (iv)Givein Part IVandtheappendices proofs forgeneral simple Liealgebras ofthefacts discovered inthepreceding sections fortheclassical ones (as, wellasonefurther important result, theWeyl character formula). Wecanatleast partially justify thisseemingly inefficient approach by saying that even ifone makes abeeline forthegeneral theorems about the structure and representation theory ofasimple Liealgebra, toapply theseresultsinpracticewewouldstillneedtocarryoutthesortofexplicit analysis oftheindividual algebras done inLectures 11~20. This is,however, afairly bald rationalization: thefactis,thereason wearedoingitthiswayisthatthis istheonlywaywehaveeverbeenabletounderstand anyofthegeneralresults. LECTURE 10 LieAlgebras inDimensions One, Two, and Three JusttogetsenseofwhataLiealgebraisandwhatgroupsmightbeassociatedtoitwewleasilyhereallLiealgebrasofdimensiontheeoles.Wewillworkprimarily with complex Liealgebras and Liegroups, but will mention thereal case aswell. ‘Needless tosay, thislecture islogically superfivous, butitieasy, fon,andserves @ Aidatic purpose, sowhy notread itanyway. The analyses ofboth theLiealgebras andtheLiegroups aecompletely elementary, with oneexception: thecasieationofthecomplexLiegroupsassociatedtoabelianLiealgebrasinvolvesthetheoryof‘complestriand shouldprobablybeskippedbyanyonenotfuiwiththissubject {10.; Dimensions one and two 4102: Dimension thee, rank one {10.3: Dimension three, rank two $104; Dimension three, rank three §10.1. Dimensions One and Two Tobegin with, anyone-dimensional Liealgebra qisclearly abelian, that is, Cwith allbrackets zero. ‘Thesimply connected Liegroup with thisLiealgebra isjustthegroup © under addition; and other connected Liegroups that have gastheir Liealgebramustallbequotientsof€bydiseretesubgroupsA.<C.fAasrank ‘one, then thequotient isjust C*under multiplication. IAhasrank two,however,Gmaybeanyoneofacontinuously varyingfamilyofcomplextoriofdimension one(otRiemannsurfacesofgenusone,orellipticcurvesoverC).‘Thesetofisomorphism classes ofsuch toriisparametrized bythecomplex plane with coordinate j,where thefunctionjonthesetoflatticesA<€is asdescribed in,eg,[AhI]. ‘Overtherealnumbers, thesituationiscompletely straightforward: theonly tealLiealgebra ofdimension one isagain Rwith trivial bracket; thesimply 134 10.LicAlgebrasinDimensions One,Two,andThree connected Liegroup associated toitisRunder addition; andtheonly otherconnected realLiegroupwiththisLiealgebraisR/Zx5? Dimension Two Herewehavetoconsidertwocases,depending onwhethergiabelianornot Case 1:9abelian, This isvery much liketheprevious case; thesimply con-nectedtwo-dimensional abeliancomplexLiegroupisjustC?underaddition, andtheremaining connected Liegroups with Liealgebra garejustquotients ofC?bydiscretesubgroups.SuchasubgroupAc€?canhaverank1,2,3, ‘or4,andweanalyzethesepossibilities inturn(thereaderwhohasseenenough‘complex(oriintheprecedingexamplemaywishtoskipdirectlytoCase2atthispoint). the rank ofAis1,wecancompletethegeneratorofAtoabasisforC*, sothatA=Ze, €Ce, ®Ce,andG&€*xC.ItherankofAis2,thereare ‘twopossibilities: eitherAfiesinaone-dimensional complexsubspaceof€? oritdoes not. Ifitdoes not, apair ofgenerators forAwillalso beabasis forC?over€,s0thatA=Ze,®Ze, C?=Ce,®Ces, andG=C*xC*. Ion theother hand Adoes lieina complex finein€?,sothat wehaveA=Ze,®Zte,forsometeEAR,thenG=Ex€willbetheproductofthetorus €(Z @Zr)andC;theremarks above apply totheclassification ofthese(seeExercise10.1).ThecaseswhereAhasrank3or4arealitlelessclear.Tobeginwith,iftherankofAis3,themainquestiontoaskiswhetheranyrank2sublatticeA’ofAliesinacomplexline.Itdoes,thenwecanassumethissublatticeissaturated (ie, apair ofgenerators forA’can becompleted toasetof generators forA)and write A=Ze,@Zxe, @Zes, sothat wewill have G=Ex C*, where Eisatorus asabove. Exercise 10.1*. Fortwoone-dimensional complex toriEand E',show that thecomplex Liegroups G=Ex€andG'=E’x€areisomorphic ifand only ifE=E*.Similarly forExC*andE’xC*. If,ontheother hand, nosuch sublattice ofAexists, thesituation ismuch ‘more mysterious. One way wecantrytorepresentGisbychoosingagenerator forAandconsidering theprojection ofC?ontothequotientofC?bythelinespanned bythisgenerator; thus, ifwewrite A=Ze,@Zea@Z(ae,+Bes) then (assuming fisnotreal)wehavemapse —— C/Ce= G=C7/Le,®Ze.Lae,+fer) —_— CZOZ) expressing Gas abundleoveratorusF=C/(Z@Z/f),withfibersisomotphic 10.1. Ditnensions OneandTwo 1s to€*.Thisexpression ofGdoesnot,however,helpusverymuchtodescribethefamily ofallsuch groups. Foronething, theelliptic curve Eissurely not determined bythedata ofG:ifwejustexchange e,ande3,forexample, wereplaceEby€/(2@Za),which,ofcourse,neednotevenbeisogenoustoE.Indeed, this yields anexample ofdifferent algebraic groups isomorphic as complex Liegroups: expressing GasaC*bundle inthisway gives itthestructureofanalgebraicvariety,which,inturn,determines theelliptic curveE(forexample,thefieldofrational functions onGwillbe the field ofrational functionsonEwithonevariableadjoined), Thus,differentexpressions ofthecomplex Liegroup GasaC*bundle yield nonisomorphic algebraic groups. Finally, thecase where Ahasrank 4remains completely mysterious ‘Among such two-dimensional complex toriaretheabelian varieties; these are justthetorithat may beembedded incomplex projective space (and hence ‘mayberealizedasalgebraicvarieties). Forpolarized abelian varieties (that is, abelian varieties with equivalence class ofembedding inprojective space) there exists areasonable moduli theory; butthesetofabelian varieties forms only acountable dense union inthesetofallcomplex tori (indeed, thegeneral complex torus possesses nononconstant meromorphic functions whatsoever). Nosatisfactory theory ofmoduli isknown forthese objects. Needless tosay,theforegoing discussion ofthevarious abelian complex Liegroups indimension two iscompletely orthogonal toourpresent pur- ‘poses. Wehope tomake thepoint, however,thateveninthisseeminglytrivial case there lurk some fairly mysterious phenomena. Ofcourse, none ofthis ‘occurs intherealcase, where thetwo-dimensional abelian simply connected tealLiegroup isjust RxRand anyother connected two-dimensional abelian tealLiegroup isthequotient ofthisbyasublattice Ac RxRofrank |or2,whichistosayeitherRx5!orS!xS'. Case2:qnotabelian.ViewingtheLiebracketasalinearmap(,3:'q—9, ‘weseethatifitisnotzero,itmusthaveone-dimensional image.Wecanthuschoose abasis {X,¥}forgasvector space with Xspanning theimage of C, }after multiplying ¥byanappropriate scalar wewillhave (X, ¥]=X, which ofcourse determines qcompletely. There isthus @unique nonabelian‘two-dimensional LiealgebragovereitherRorC.‘WhatarethecomplexLiegroupswithLiealgebra4?Tofindone,westartWiththeadjointrepresentation ofg,whichisfaithful:wehave ad(X): X40, ad(¥): XH —X, Yoox, yeso orinmatrix notation, interms ofthebasis {X,¥}for9, or -10 attheadjoint forrm 136 10,LieAlgebrasinDimensions One,Two,andThree ‘ab Topologically thisgroup ishomeomorphic to€xC*.Totake itsuniversalcover,wewriteageneralmemberofGyas es o ay ‘The product oftwosuch matrices isgiven by ets\fe"s\_ (estes o lo Jo 1 sowemay realize theuniversal cover GofGyasthegroup ofpairs (t,3) €xwith group law (Ws) HU EsHes, ‘ThecenterofGisjustthesubgroup Z(G)=(Qnin,0)}=Z, sothat theconnected groups with Liealgebragformapartiallyorderedtower G 1 4 G,=Gin ={(a,byeC*xC5(0,B)-(a’,b’) =(aa’, b+a°b’)}. 1 J Gy Exercise 10.2%. Show that fornm thetwo groups G,and G,,arenot isomorphic. Finally, inthereal case things aresimpler: when weexponentiate the adjoint representation asabove, theLiegroup wearrive atisalready simply connected, andsoistheuniqueconnectedrealLiegroupwiththisLiealgebra. §10.2. Dimension Three, Rank 1 {Asinthecaseofdimension two,welookattheLiebracketasalinearmapfrom‘qogandbeginourclassification byconsidering therank ofthis map (that is,thedimension of%q), which may beeither 0,1,2,or3.Forthecase $102. Dimension Thee, Rank 1 37 ofrank 0,werefer back tothediscussion ofabelian Liegroups above. We begin with thecase ofrank I. Here thekernelofthemap[ ,]:A?g -+giswodimensional, which meansthatforsomeX€qitconsistsofalvectorsoftheformX Ywith¥ranging. ‘overall ofg(Xherewilljustbethevector corresponding tothehyperplane ker({, J)<Agunder thenatural (uptoscalars) duality between athree- dimensional vector space anditsexterior square). Completing Xtoabasis,{X,¥,Z}ofg,wecanwriteintheform (4 Y1= (4,Z1=0, (42)=0x4BYyz forsomea, f,€ €.Ifeither foryisnonzero, wemay now rechoose ourbasi, replacing ¥byamultiple ofthelinear combination aX+PY+yZandeither leaving Zalone (iff40)orreplacingZby¥(if¥0).Wewillthenhave. (X, ¥1=%,2]=0, 21-7 fromwhichweseethatqisjusttheproductoftheone-dimensional abelianLiealgebra CXwith thenon-abelian two-dimensional Liealgebra CY@CZ described inthepreceding discussion. Wemay thus ignore thiscase and assume that infactwehave fi=y=0;replacing XbyaXwethen have the Liealgebra (%Y1=0%Z1=0, z21=x. How doWwefind theLiegroups with this Liealgebra? Asbefore, weneedtostartwithafaithfulrepresentation ofg,butheretheadjointrepresentation‘isuseless,sinceXisinitskernel.Wecan,however,arriveatarepresentationofqbyconsidering theequations defining g:wewant tofind apair of endomorphisms ¥andZonsome vector space that donotcommute, butthatdocommutewiththeircommutator X=[¥,Z];thus, (YZ~ZY)—(¥Z—ZY)¥ =YZ—2¥ZY+ZY?=0 andsimilarly for[Z,CY,Z1]. One simple way tofind such apair ofendo-morphisms ismakealltreetermsY?Z,YZY,andZ?Yintheaboveequation2210, eg,bymaking Yand Zboth have square zero, and tohave YZ=0 while ZY¥0.Forexample, onathree-dimensional vector space with basis ¢,.¢2,and eywecould take ¥tobethemap carrying ¢,toe,and killing ¢and e,,andZthemap carrying e,to¢,andkilling ¢,andes;wethen haveYZ=OwhileZYsendseytoe,.WeseethenthatgisusttheLiealgebranyofstrictlyupper-triangular 3x3matrices.Whenweexponentiate wearrivetthe group, 138 10.LieAlgebrasinDimensions One,Two,andThree tae G=t{{0 1chabeee oot which issimply connected. Now thecenter ofGisthesubgroup to 6) z@=4{0 1olvech xe, oo sothediscretesubgroups ofZ(G)arejustlatticesAofrank1or2;thusanyconnected group with Liealgebra giseither G,G/Z, orG/(2 xZ)—that is, an extension of€x€byeither€,C*,oratorusE. Exercise 10.3. Show that G/A isdetermined uptoisomorphism bytheone- dimensional Z(GY/A Asimilaranalysisholdsintherealcase:justasbefore,yistheuniquerealLiealgebraofdimension three with commutator subalgebra ofdimension one,itssimplyconnected formisthegroupGofunipotent 3x3matricesand(thecenterofthisgroupbeingR)theonlyothergroupwiththisLiealgebraisthequotient H=G/2. Incidentally, thegroup Hrepresents aninteresting example ofagroup that cannot berealized asamatrix group, ic,that admits nofaithful finite- dimensional representations. One way (oseethis istoargue that inany irreducible finite-dimensional representation Vthecenter 5?ofIH,beingcompactandabelian,mustbediagonalizable; andsounderthecorresponding,representation oftheLiealgebra gtheelement Xmust becarried toa iagonalizable endomorphism of¥.Botnow ifv€VisanyeigenvectorforX with eigenvalue 2,wealso have, arguing asin§9.2, X(¥(0)=YEX(W)=Y(A0)=AY(0) andsimitarly X(Z(0)) =2Z(0), ic,both ¥(e)andZ(o) arealso eigenvectors forXwith eigenvalue 2.Since ¥and Zgenerate gand therepresentation V isirreducible, itfollows thatXmust actasascalar multiple2:1oftheidentity, butsinceX=[¥,ZJisacommutator and50has(race0,itfollowsthat2=0. Exercise 10.4*, Show that ifGisasimply connected Liegroup, anditsLie algebra issolvable, then Gcannot contain anynontrivial compact subgroup{inparticular, itcontainsnoelementsoffiniteorder), ‘The group Hdoes, however, have animportant infinite-dimensional repre- sentation. This arises from therepresentation oftheLiealgebra gonthespace ¥of6functionsonthereallineRwithcoordinatex,inwhich¥,Z,andX aretheoperators $103. DimensionThee,Rank2 be» Vifrente: f spat25M i andX=CY,ZJis ~xitimes theidentity. Exponentiating, weseethate"”acts ‘ona function fbymultiplying itbythefunction (cos¢x+i-sin tx};€sends JStothefunction F,where F(x) =f(t+x), and esends ftothescalar multiple en" f. §10.3. Dimension Three, Rank 2 Inthiscase, write thecommutator subalgebra @q. qasthespan of(wo elements Yand Z.The commutator ofYand Zcan then bewritten (YZ) =aY+pz. Butnowtheendomorphism ad(¥)ofgcarriesgintoMq,killsY,andsendsZtoaY +AZ,and sohastrace f;ontheother hand, since ad(Y)is acommutator inEnd{(q),itmusthavetrace0.Thus,f,andsimilarly a,mustbezero;ie.,thesubalgebra Pqmustbeabelian.ItfollowsfromthisthatforanyelementX€g notinPq, themap ad(X): Pq >Fg ‘must beanisomorphism. Wemay now distinguish two possibilities: either a1(X) isdiagonalzible oritnot (Note that forthefirst time weseeacase where theclassification ofthe realLiealgebra willbemorecomplicated thanthatofthecomplex: intherealcasewewillhave todeal with thethird possibility thatad(X) isdiagonalizible ‘over €butnotover R,ic,that ithastwocomplex conjugate eigenvalues. Though wehave notseen itmuch inthese low-dimensional examples, infact itisgenerally thecase thattherealpicture issubstantially more complicated than thecomplex one, foressentially justthisreason.) Possibility A:ad(X) isdiagonalizable, Inthiscase itisnatural touseasabasis forga pair ofeigenvectors ¥,Zforad(X),and bymultiplying Xbyasuitable scalar wecanassume that oneoftheeigenvalues (both ofwhich arenonzero) isJ,Wethushavetheequations forg (KYYI=% W%Z)<az, z1=0 cos) forsome aeC*. Exercise 106,Show that woLicalgebrasg,, 0,corresponding totwodiferentsealarsinthestructureequations(103)areisomorphic ifandonlyif2=aor 40 10.LieAlgebrasinDimensions One,Two,andThree a=Ia’.Observethatwehaveforthefirsttimeacontinuously varyingfamilyofnonisomorphic complexLiealgebras. Tofindthegroups with these Liealgebras wegototheadjoint represen- tation, which here isfaithful. Explicitly, ad(¥) carries Xto—¥and kills ¥ andZ;ad(Z) carries Xto—aZ andalso kills ¥andZ;andad(X) carties ¥ toitself, Ztoa2,and kills X.Ageneral member aX~bY~eZoftheLie algebra isthus represented (with respect tothebasis {Y,Z,X}forg)bythe matrix a0 6 0 aa ac 000 Exponentiating, wefindthatagroupwithLiealgebra9is eo Ww G= jo&* o),uuvec} GLC oot Here werunacross avery interesting circumstance. Ithecomplex number « isnot rational, then theexponential map from qtoGisone-to-one, andhence ahomeomorphism; thus, inparticular, Gissimply connected. If,ontheother hand, «isrational, Gwillhave nontrivial fundamental group. Toseethis,“observethatwealwayshaveanexactsequenceofgroups 14B4G+A-1, where a) A=4lo et Ole oot and bow B= 1/01o).noe€} xOxe oot Now when aQ,thegroup A=Cissimply connected; butwhen a Q— ‘whateveritsdenominator—we haveA%C*andcorrespondingly x(G)=Z. Exercise 10.7. Show that Ghasnocenter, and hence when a#Q,itis the ‘unique connected group with Liealgebra g.For« Q,describe theuniversal covering ofGand classify allgroups with Liealgebra g, ‘Observe that inthis case, even though wehave acontinuously varying family ofLiealgebras a,,wehave nocorresponding continuously varying $104. DimensionThree,Rank3 “1 family oftheadjoint (linear) Liegroups; thesimply-connected forms doform afamily, however. Possibility B:ad(X) isnot diagonalizable, Inthiscase thenatural thingtodo istochoose abasis {¥,Z}of9gwith respect towhich ad(X) isinJordan ‘normal form; replacing Xbyamultiple, wemay assume both itseigenvalues are1sothat wewill have theLiealgebra (WY=% (Zs +z, (42-0 (108) ‘With respect tothebasis (YZ, X}for9,then,theadjointactionofthe general clement aX—bY—cZoftheLiealgebra isrepresented bythematrix aa bte Oa ¢ 00 0 andexponentiating wefindthat thecorresponding group is el te w G={{0 e vluuoeeh. \o o 1 Exercise 10.9. Show that thisgroup hasnocenter, and hence istheunique connected complex Liegroup with itsLiealgebra. NotethattherealLiegroupsobtainedbyexponentiating theadjointaction ‘oftheLiealgebrasgivenby(10.5)and(10.8)areallhomeomorphic toR?andhave nocenter, andsoaretheonly connected realLiegroups with these Lie algebras, Exercise 10.10. Complete theanalysis ofreal Liegroups inCase 2bycon- sidering thethird possibility mentioned above: that ad(X) acts on2gwith distinct complex conjugate eigenvalues. Observe that inthisway wearrive atourfirst example oftwo nonisomorphic real Liealgebras whose tensorProductswith€areisomorphic. §10.4. Dimension Three, Rank 3 Ovranalysisofthisfinalcasebegins,asintheprecedingone,bylookingfortigenvectors oftheadjointactionofasuitableelementX€g.Specifically, wetlaim that wecan find anelement Heq such that ad(H):q—9 hasan igenvector with nonzero cigenvalue. Toseethis, observe fist that forany nonzero X€4, therank ofad(X) must be2;inparticular, wemust have Kerfad(X)) =CX. Now start with anyX€9.Eitherad(X)hasaneigenvector Withnonzero eigenvalue oritisnilpotent: ifitisnilpotent, then there exists a 1 10.LieAlgebras inDimensions One,Two,andThree vector ¥€4,notinthekernel ofad(X) butinthekernel ofad(X)?—that is,such that ad(X)(¥) =aX forsome nonzero aeC.But then ofcourse ad(¥)(X) =—aX, sothat Xisaneigenvector forad(Y) with nonzero eigenvalue, ‘So: choose HandX€gothatXisaneigenvectorwithnonzeroeigenvalue forad(H), andwrite [H,X]=aX.Since H€9,ad(H) isacommutator in End(g), and sohastrace 0;itfollows that ad(H) must have athird eigenvector ¥with eigenvalue —a. Todescribe thestructure ofgcompletely itnow remains tofind the commutatorofXandY;butthisfollowsfromtheJacobi identity. We have (CH,0%,YT)=—04,04 01~1%1H,XT =-[X,a¥] -(%ax] =o, from which wededuce that [X,¥]must beamultipleofH;sinceitmustbe anonzeromultiple,wecanmultiplyXor¥byascalartomakeitI.Similarly multiplyingHfbyascalarwecanassumeais1oranyothernonzeroscalar Thus,thereisonlyonepossiblecomplexLiealgebra9ofthis type. One could look forendomorphisms #1,X,and Ywhose commutators satisfy these relations,aswedidbefore.Orwemaysimplyrealizethatthethree-dimensional Liealgebra o1;€ hasnotyetbeen seen, soitmust bethislastpossibility. In fact, anatural basis forst3€ is 10 1 0 0)n-()2).x-(5 re8) whose Liealgebra isgiven by ULxXJ=2x, (HYJ=-2y (XYJ=m (lor What groups other than SLE have Liealgebra sl,€? Tobegin with, the group SLCis simply connected: forexample, themap SLC -+€?~{(0,0)} sending amatrix toitsfirst rowexpresses thetopological space SL3€ as@ ‘bundle withfiber €over€?—{(0,0)}. Also, itisnothard toseethatthecenter ‘ofSLC isjust thesubgroup {$1} ofscalar matrices, sothat theonly other connected group with Liealgebra st3€ isthequotient PSL,C =SI,€/{ +1}. ‘Asin theprecedingcase,theanalysisofrealthree-dimensional Liealgebras 4with 9q=ginvolves oneadditional possibility. Attheoutset oftheargu- iment above, westarted with anarbitrary H€gand said that ifad(Z4) hadnoeigenvector otherthanHitself,thenitwouldhavetobenilpotent. Ofcourse,intherealcaseiisalsopossiblethatad(H)hastwodistinctcomplexconjugateeigenvalues AandZSince ad(H) isacommutator inEnd(q) andsohastrace 0,Awill have tobepurely imaginary inthiscase; and somultiplying HTbya real scalar wecan assume that itseigenvalues areiand—i.Itfollowsthen thatwecanfindX,Yewith $103. Dimension Three, Rank 2 143 (HX1=¥ and [HY]=—X. UsingtheJacobiidentityasbeforewemayconcludethatthecommutator ofXandYisamultipleofH;aftermultiplyingeachofXandYbyarealscalar wwecan assume that itiseither Hor—H. Finally, if[X,Y] =—H, then weobserve that weareinthecase weconsidered before: ad(¥) will haveX+Hasaneigenvector withnonzeroeigenvalue, andfollowingourpreviousanalysis wemay conclude that 9=<l,R. Thus, weareleftwith thesole additional possibility that ghasstructure equations (x=% CYI=-x, (YI=H (10.12) ‘This, finally, wemay recognize astheLiealgebra sit,ofthereal Liegroup SUG) (asyoumay recall, theisomorphism su, @€&s1,C wasused inthe last lecture).‘WhataretherealLiegroupswithLiealgebrass1,Randsu,?Tostart,the ‘enter ofthegroup SLR isagain justthescalar matrices (+1}, sotheonlygroupdominated bySL-yRisthequotientPSL4R.Ontheotherhand,unlikethecomplex case SLR isnotsimply connected: now themap associating to 2x 2matrix itsfirst row expresses SLR asabundle with fiber Rover R?—((0,0)}, $0thatx,(SL,R) =Z.More precisely PSLgR maps tothereal projective lineP'R, which ishomeomorphic tothecircle, with fiber homeo-morphictoR2,0x,(PSL,R) =Z.Wethushaveatowerofcoveringspacesof PSL,R, consisting ofthesimply-connected group 5withcenter Zandits quotients 5,=S/nZ(notalofthese arecovers ofSLaR, despite thediagram below). Anote: In§102 weencountered areal Liegroup with nofaithful finite- dimensional representations; only itsuniversal cover could berepresented as ‘amatrix group. Here wefindinsome sense theopposite phenomenon: thegroups Sand 5,havenofaithful finite-dimensional representations, allfinite- dimensional representations factoring through SLR orPSLR. This factwillbeprovedasaconsequence ofourdiscussion oftherepresentations oftheLiealgebra sly inthenext lecture. What about groups with Liealgebra stiz? Tobegin with, there isSU(2}, which (again viathemap sending amatrix toitsfirst rowvector) ishomeo- morphic toS®and thus simply connected. The center ofthisgroup isagain {41}, s0that thequotient PSUQ) istheonly other group with Liealgebrasu,.Alternatively, wemayrealizeSU(2)asthegroupofunitquaternions. cf.Exercise 7.15) Finally, weremark that there areother representations ofthereal and complex Liegroups discussed above. Aswewillsee,theLiealgebra s0€ is isomorphic tos13C, which induces anisomorphism between thecorrespondingadjointformsPSL,€andSO,€(andbetweenthesimply-connected forms SLC andthespin group SpingC). This inturn suggests twomiore realforms ofthis group: SOR andSO*2,1),Infact,itisnothardtoseethatSO,R= PSUQ2), while SO*Q, 1)=PSLAR. Lastly theisomorphism suy,, OC = “4 10.LieAlgebrasinDimensions One,Two,andThree ‘su,@€=sl,CimpliesthattherealLiealgebrast,,isisomorphic toeithersui,oFslg; infact,thelater isthecaseandthisinduces anisomorphism ofgroupsSU;,,sSLR.Wesummarize theisomorphisms mentioned inthediagram below: 5 & Sping€ SpinR SU(I,1) =SLR c__, SL,€ «> SUQ) ={unit quaternions) PSL,R c. PSLy€ «> PSUQ2) =(unit quaternions}/+1 S0'Q, Ye, SOC > SOR (10.13) Note also the coincidences: Spx{€)=SL{O, —_Sp,(R) =SLR), (10.14) which follow from thefact that Sprefers topreserving askew-symmetric bilinear form, and for2x2matricesthedeterminant issuchaform. Exercise10.15.ldentilytheLiealgebras505,sitysuy,1,$02,1, andverifytheassertions madeaboutthecorresponding Liegroupsinthediagram. Exercise 10.16. For each oftheLiealgebras encountered inthis lecture, ‘compute thelower central series andthederived series, andsaywhether the algebra isnilpotent, solvable, simple, orsemisimple. Exercise 10.17. The following areLiegroups ofdimension two orthree, s0 ‘must appear onourlist.Find them: ()thegroup ofaffine transformationsof theline(x+-+ax +b,under composition); (i)thegroup ofupper-triangular 2x2imatrices (i)thegroupoforientationpreservingEuclideantransforma- tionsofthe plane (compositions oftranslations androtations). Exercise 10.18, Locate R?with theusual cross-product onourlistofLie*algebras,Moregenerally, considerthefamilyofLiealgebrasparametrized by real quadruples (a,b,c, d),each with basis X,¥,Zwith bracket given by LYY)=aZ4d¥, (%Z]= bX,[ZX]=e¥—az. $104. Dimension Three, Rank 3 4s, Classify thisLiealgebra as(a,b,c, d)varies inR*,showing inparticular that ‘every three-dimensional Liealgebra canbewritten inthisway. Exercise 10.19. Realize theisomorphism ofSU(L, 1)with SLR byidentifying them with thegroups ofcomplex automorphisms oftheunit disk and the upper half-plane, respectively. Exercise 10.20, Classify allLiealgebras ofdimension four and rank 1;inparticular, showthattheyarealldirectsumsofLiealgebrasdescribedabove. Exercise10.21,ShowmoregenerallythatthereexistsaLiealgebraofdimen-sionmandrank1thatisnotadirectsumofsmallerLiealgebrasifandonlyifmisodd; incase misodd show that thisLiealgebra isunique andrealize itas.a Liesubalgebra ofs,C. LECTUPE IL Representations ofsl,C This isthe frstoffour lectures—S11--14—that comprise insome sense theheat of thebook. Inparticular, thenaive analysis of 11.1,tether with teanalogous pats(of12and613formthepareigmforthestdyofitedimensional representations‘ofallsemisimpleLiealgebrasandgroups.611.28lescentralinitweshowhowtheanalysis carried outin§11.1 canbeused toexplicitly describe thetensor products of irreducible representations. 6113s last important; itindicates howwecaninterpret, seometricallysomeoftheresultsoftheprecedingsection.Thediscussionsin§1.1and S112arecompletelyelementary(wedovsethenotionofsymmeitipowersofvector scebutinnon-threatening way)§113involvesa fairamountofelassicalprojectivescometry, and canbeskimmed orskipped bythose notalrendy familiar with the ‘elevant basi notions from algebraic geometry. 11.1:Theiereducblerepresentations$112: (tie plethysm 11.3: Aie peometre plethysm §1L.1. The Irreducible Representations Westartourdiscussionofrepresentations ofsemisimple Liealgebras with hesimplestcase,thatofs13€.Aswewillsee,whilethiscasedoesnotexhibitany ofthecomplexity ofthemore general case, thebasic idea that informs thewholeapproach isclearlyillustrated here. ‘This approach isone already mentioned above, inconnection with the representations ofthesymmetric grouponthree letters. The idea inthat case was that given arepresentation ofour group onavector space Vwefist restrict therepretentation totheabelian subgroup generated byaJcyde Weobiain adecomposition S11. TheIreducble Representations “7 v=O% ofVintoeigenspaces fortheaction oft;thecommutation relations satisfiedbytheremaining elementsoofthegroupwithrespectto«impliedthatsucho simply permuted these subspaces V,,sothattherepresentation wasineffect determinedbythecollectionofeigenvalues oft. ‘Ofcourse,circumstances inthecaseofLiealgebrarepresentations arequitedifferent: tonametwo,itisnolongerthecasethattheactionofan abelian object onanyvector space admits such adecomposition; andeven ifsuch a ‘decomposition exists wecertainly cannot expect thattheremaining elements ofourLiealgebra willsimply permute itssummands. Nevertheless, theidearemainsessentially agoodone,asweshallnowsee.Tobegin with, wechoose thebasis fortheLiealgebra sl, from thelast lecture: 10} 01 0 0a=(5i).x-(obr-((°) satisfying (H,X)=2x, (H,¥J=-2Y, (X,¥]=H. (ily These seem likeaperfectly natural basis tochoose, butinfactthechoice is dictated bymore than aesthetics; there is,aswewillsee,anearly canonical wayofchoosing abasisofasemisimple Liealgebra(uptoconjugation), which wiltyield thisbasis inthepresent circumstance and which will share many of theproperties wedescribe below. Tnanyevent, letVbeanirreducible fnite-dimensional representation of alyC. Westart bytrotting outoneofthefacts that wequoted inLecture 9, thepreservation ofJordan decomposition; inthepresent circumstances it implies that The action ofHtonVisdiagonatizable. (nay ‘Wethus have, asindicated, adecomposition v=Ohe (11.3) wherethearunovercollectionofcomplexnumbers,suchthatforanyvectorveV,wehave Heo) =ev. Thenextquestion isobviously howXand¥actonthevariousspaces¥,.We claim thatXandYmust each carry thesubspaces V,intoother sub- afaces ¥,.Infact, wecan bemote specific ifwewant toknow where the ‘image ofagiven vector v€V,under theaction ofXsitsinrelation tothe ‘ecomposition (11.3, wehave toknow how Hacts onX(o this isgiven by the us 11,Representations ofs1,€ Fundamental Calculation (frst time H(X() =X(H(0) +CH,X10) =X (a0)+2X(o) = 2-X(0} iifvisan eigenvector forHwitheigenvaluea,thenX(o)isalsoaneigenvector JorH,with eigenvalue &+2.Inother words, wehave XiV+ Vesa ‘The action of¥oneach ¥,issimilarly calculated; wehave ¥(V,) <Va Observe that asanimmediate consequence ofthis and theirreducibility of ¥,allchecomplex numbers #that appear inthedecomposition (11.3) must be congruent tone another mod 2:for any thatactually occurs, thesubspace DBVoutan would beinvariant under sl,€ and hence equal (oallof¥.Moreover, bythe same token, the¥,thatappear must form anunbroken string ofnumbers oftheformfl+2,...f!-+2k.Wedenote bynthelastelementinthissequence;althispoint wejustknow nisacomplex number, butwewillsoon seethat itmust beaninteger. Toproceedwithouranalysis,wehavethefollowingpictureoftheaction ofsl,€ onthevector space ¥: x x x i a a Q Qa Choose anynonzero vector v€¥,;Since Vyyy =(0),wemust have X(0) =0. ‘Weasknowwhat happens when weapply themap Ytothevector v.Tobegin with, wehave Claim 11.4,Thevectors (0,Y(0), Y(0),..-} span V. Proor. From theirreducibility of¥itisenough toshow that thesubspaceWcVspannedbythesevectorsiscarriedintoitselfundertheactionofs,€. Clearly, ¥preserves W,since itsimply carties thevector ¥™(o) into Y"""(0, Likewise, since thevector Y™(0) isinV,-amy Wehave 14(Y™()) =(n—2m): ¥*(o),80 HpreservesthesubspaceW.Thus,itsufficestocheckthatX(W)< Wie,thatforeachm,Xcarties¥"(o)intoalinearcombination ofthe¥"(o. ‘Wecheck thisinturn form=0,1,2, ete: Tobegin with, wehave X(o) =0€W.Toseewhat Xdoes to¥(o), weuse S111. TheIrreducible Representations 9 thecommutation relations forsls: wehave X(V(O)=LXVIO)+XC) =Heo) +YO) Next, weseethat XC)=OX,YIN) +FOX) =HY)+Yorn) =(n~2)- Yo) +m YOO) ‘Thepattern now isclear: Xcarries each vectorinthesequencev,¥(o)¥%(0), intoamultipleofthepreviousvector.Explicitly, wehave X(VMO))=(1(2)+(=A=Im+2)YHHOY, or X(YO)=mn=m+1YO), ans ascanreadily beverified byinduction. o There areanumber ofcorollaries ofthe calculation intheabove Claim. Tobegin with, wemake theobservation that alltheeigenspaces V,ofHareonedimensional. (11.6) Second, since wehave inthecourse oftheproof written down abasis forV and said exactly where each ofH,X,and Ytakes each basis vector, the representation Viscompletely determined bytheonecomplex number nthat westarted with; inparticular, ofcourse, wehave that Visdetermined bythecollectionofaoccurringinthedecompositionva@ye. (17 Tocomplete ouranalysis, wehave touseone more time thefinite dimen- sonality of¥.This tells usthat there isalower bound ontheaforwhich ¥,#(O)aswell asanupper one,sothatwemust have ¥*(v) =Ofor suicently large k.Butnow ifmisthesmallest power of¥annihilating o,then from the ‘lation (11.5), = X(YM(O))=min=m+IYHO), ‘ndthefactthat Y"-1(0) #0,weconclude that n~m-+ 1=0;inparticular, itfollows thatm&anon-negative integer. Thepicture isthusthattheeigen- valuesaofHon¥formastringofintegersdifferingby2andsymmetricabout theorigininZ.Insum,then,weseethatthereisauniquerepresentation V°foreach non-negative integerm;therepresentation ¥is(n+1)-dimensional, withHhaving eigenvalues n,n~2,....—n+ 2—n. 150 11,Representations ofsl, Notethattheexistencepartofthisstatement maybededucedbycheckingthattheactionsofH,X,and¥asgivenaboveintermsofthe basis 0,Yo,¥%(0),..-,¥"(o)for¥doindeedsatisfyallthecommutation relationsforst;C.Alternatively, wewill exhibit them inamoment. Note also that bythesymmetry ofthe eigenvalues wemay deduce theuseful factthatanyrepresen- tation VofslyC such that theeigenvalues ofHallhave thesame parity andoccurwithmultiplicity oneisnecessarly irreducible;moregenerally,thenumber ofirreduciblefactorsinanarbitraryrepresentation Vofsl€isexactlythesum ofthemultiplicities of0and 1aseigenvalues ofH. ‘Wecanidentifyinthesetermssomeofthestandardrepresentations ofst,C. Tobegin with, thetrivial one-dimensional representationCsclearlyjustV. Asforthestandardrepresentation ofsl,onV=C?,ifxandyarethestandard basis forC2,then wehave H(x) =xandH(y) =—y,sothat V=Cx®C-y=V_,@M;isjusttherepresentation V"above.Similarly,abasis forthesymmetricsquareW=Sym?=Sym?C?isgivenby{x3,xy,y*}and wehave (xx)=xHG)+MG)=Dex, H(ey)=xH(y)+H(x)-y=0, H-y) =HQ) +HOY=~2y-y sotherepresentation W=C-x? ®C-xy@C-y? =W.2@ Wo@ Wyisthe representation ¥°above. More generally, thenthsymmetric power Sym*V ofVhasbasis(x",x*""y,...,y"},andwehave Hey) =(9B HG) ey eH) yt a(n exttyt +0thattheeigenvalues ofHonSym*V areexactly n,n~2,..., —m.Bythe observation above that arepresentation forwhich alleigenvalues ofHoccur with multiplicity 1must beirreducible, itfollows that Sym*V isirreducible, and hence that VO=Sym*V. Insum then, wecansaysimply that “Anyirreducible representation ofsl,Cisasymmetric powerofthestandardrepresentation V=C2 (18) Observe thatwhen weexponentiate theimageofsCundertheembedding s1,C +sl,,,€ corresponding totherepresentation Sym'V, wearrive atthegroupSL,€whenmisoddandPGL3Cwhenniseven,Thus,therepresentationsofthegroupPGLCareexactlytheevenpowersSym?V. Exerelse 11.9. Usetheanalysis oftherepresentations ofsl,€ toprove the statement made intheprevious lecture thattheuniversalcoverSofSLRhes nofinite-dimensional representations $11.2 ALittle Pethysm ist §11.2. ALittle Plethysm Clearly,knowingtheeigenspace decomposition ofgiven representations tellsustheeigenspace decomposition ofalltheirtensor,symmetric, andalternatingproductsandpowers:forexample, ifV=@)¥,andW=(DW,thenV@W= QW,@Wp)andV,@WisaneigenspaceforHwitheigenvaluew+f.We canusethistodescribe thedecomposition ofthese products andpowers into irreducible representations ofthealgebra s1,. Forexample, letV=C?bethestandard representation ofs1,C; and suppose wewant (ostudy therepresentation Sym?¥ ®Sym?V; weaskin particular whether ifitirreducible and, ifnot, how itdecomposes. Wehave seenthattheeigenvalues ofSym?V are2,0,and—2,andthose ofSym?V are3,1,~1,and—3.The12eigenvalues ofthetensorproductSym?V@Sym?V arethus 5and —5,3and —3(taken twice), andIand —1(taken three times), wwemay represent them bythediagram —*-}+-©+©+©+-O-++-— s ° 5 Theeigenvector with eigenvalue 5willgenerate asubrepresentation ofthe tensor product isomorphic toSym*¥, which willaccount foroneoccurrenceofeachoftheeigenvalues 5,3,1,~1,~3,and~5.Similarly,thecomplementofSym*Vinthetensorproductwillhaveeigenvalues 3and—3,and!and=I(taken twice), andsowillcontain acopy oftherepresentation Sym?¥, which willaccount foroneoccurrence oftheeigenvalues 3,1,—1and —3;andthecomplement ofthesetwosubrepresentations willbesimplyacopyofV.Wehave, thus, Sym?V@Sym?=Sym*V@Sym?V@¥. Note thattheprojection map Sym?V @Sym?-+Sym'V(onthefirstfactorisjustmultiplication ofpolynomials; theothertwoprojec-tions donotadmit such obvious interpretations. Exercise 11.10. Find, inasimilar way, thedecomposition ofthe tensor product Sym'V @Sym*¥. Exercise 11.11%, Show, ingeneral, thatfora>hwehave Sym*V @Sym*V =Sym**V@Sym*"*-?V @-~@Sym*-*¥. Asindicated, wecanalsolookatsymmetric andextetiorpowersofgivenrepresentations; inmany ways this ismore interesting For example, let 182 11Representations ofly V=C*beasabovethestandardrepresentation ofs€,andletW=Sym?Vbeitssymmetric square; ie,inthenotation introduced above, take W=V®.Weasknowwhetherthesymmetric squareofWisireducible, andifnotwhatitsdecomposition is.Toanswer this,observe thatWhaseigenvalues ~2,0, and 2,each occurring once, sothat thesymmetric square ofWwill haveeigenvalues thepairwisesumsofthesenumbers—that is,—4,~2,0(occurring,tovice), 2,and4.Wemayrepresent Sym? bythediagram: —e—_}—*—}-@—} eo43 2 4 0 1 23 4 From this,itisclear thattherepresentation Sym? must decompose into ‘onecopy oftherepresentation V“=Sym*V, plusonecopy ofthetrivial (one-dimensional) representation: Sym?(Sym?)) =Sym*v @Sym°Y, (nig) Indeed,wecanseethisdirectly:wehaveanaturalmap, ‘Symm?(Sym?V)) +Sym4¥ ‘obtained simply byevaluation; thiswillhave aone-dimensional kernel (ifxandyareasabovethestandardbasisforVwecanwriteageneratorofthiskernel as(x?)-(y?) —(xy), Exercise (1.13. Show that theexterior square AWisisomorphictoWitselt. ‘Observe thatthis,together withtheabove description ofSym?W, agrees with thedecomposition ofW@Wgiven inExercise 11.11 above. Wecan,inasimilar way, describe thedecomposition ofallthesymmetric powers ofthe representation W=Sym?¥: Forexample, thethird syminetricpowerSym?Whaseigenvalues givenbythetriplesumsoftheset{—2,0,2};these are ~6, ~4, ~2(twice),0(twice),2(twice),4,and6;diagrammatically, th tO OO ‘Again, there isnoambiguity about thedecomposition; this collection ofeigenspaces canonlycomefromthedirectsumofSym®VwithSym?sowemust have ‘Sym?(Sym?)=Sym®V@Sym?¥.Asbefore,wecanseealeastpartofthisdirectly:wehaveanaturalevaluationmap Sym?(Sym?V)-» Sym°V, 11.3. ALittle Geometric Plethysm 153 andtheeigenspace decomposition tells usthat thekernel istheirreducible representation Sym?¥. Exercise 11.14. Usetheeigenspace decomposition (oestablish theformula twat foralln §11.3. ALittle Geometric Plethysm Wewant togivesome geometric interpretations ofthese andsimilar decom-positionsofhighertensorpowersfrepresentations ofs,C.Onebigdilferenceisthat instead oflooking attheaction ofeither theLiealgebra sl,€ orthegroupsSL,CorPGL,Conarepresentation IY,welookattheactionofthe sroup PGL,C ontheassociated projective space' PW. Inthiscontext, iis natural tolook atvarious geometric objects associated totheaction: for ‘example, welook atclosures oforbits oftheaction, which allturn outtobe algebraic varieties, ie,definable bypolynomial equations. Inparticular, our goalinthefollowing willbetodescribe thesymmetric andexterior powers of, WintermsoftheactionofPGL,ContheprojectivespacesPWVandvarious foci inPW, ‘Themain point isthat while theaction ofPGL,€ ontheprojective space PY=P*associatedtothestandardrepresentation Vistransitive,itsaction ‘onthespaces P(Sym"V) &P*forn >1isnot.Rather, theaction willpreserve various orbits whose closures arealgebraic subvarieties ofP*—for example, thelocus ofpoints C= {{o-0-...:0]:0€ Y)<P(Sym*¥) corresponding tonth powers inSym"V will beanalgebraic curve inP(sym"V) =P*,calledtherationalnormalcurve;andthiscurvewillbecarriedintoitself byanyelement ofPGL,C acting onP*(more about thisina moment). Thus,aknowledgeofthegeometryofthesesubvarieties ofPWmay illuminatetherepresentation W,andviceversa.Thisapproachisparticularly Useful indescribing thesymmetric powers ofW,since these powers canbe viewed asthevector spaces ofhomogeneous polynomials ontheprojective space P(W*) (or,mod scalars, ashypersurfaces inthat projective space). Decomposing these symmetric powers should therefore correspond tosome interesting projective geometry. "PWherdenotestheprojectivespaceoftinesthroughtheorigininW.oFthequotientspaceof {\(0}bymoltptication bynonzeroveslarswewrite[]forhepontinPIVdeterminedbythe sontvectorw.ForWmO°"[2s2a]ithepntinB=PWdeterminedbyapoint Fysovtain" 154 1,Representations ofsly Digression onProjective Geometry First, aswehave indicated, wewant todescribe representations ofLiegroupsintermsofthe corresponding actions onprojective spaces. Thefollowing factromalgebraicgeometryisthereforeofsomemoralifnotlogical)importance: Fact 11.15. Thegroup ofautomorphisms ofprojective space P"—cither asalgebraicvarietyorascomplexmanifold—is justthegroupPGL,4,€. Foraproof,see(Hal.(FortheRiemannsphereP'atleast,thisshouldbe 4familiar factfrom complex analysis) Foranyvector space Wofdimension n+1,Sym'W* isthespace of homogeneous polynomials ofdegree kontheprojective space P*=PW oflinesinW;dually,Sym*Wwillbethespaceofhomogeneous polynomials ofdegreekontheprojective spaceP*=P(W*)oflinesinW*,orofhyperplanes inW.Thus, theprojective space P(Sym'¥) isthespace ofhypersurfaces of degree kinP*=P(W*) (Because ofthisduality, weusually work with objectsintheprojectivespaceP(W*)rather thanthedualspacePWinordertoderiveresults about symmetric powers SymtW; thismay seem initially more con- fusing, butwebelieve itisultimately lesss0) Foranyvector space Vandanypositive integer», wehave anatural map, called theVeronese embedding PV coP(Sym"V*) thatmapsthelinespannedbyo€¥*tothelinespannedbyveSym"V*,Wewillencounter theVeronese embedding ofhigher-dimensional vector spaces inlater lectures; here weareconcerned just with thecase where Vistwo dimensional, soPY* =P". Inthiscase wehave amap 1cP! SsPt=PSym"V*) whoseimageiscalledtherationalnormalcurveC=C,ofdegreen.Choosing bases {af} forV*and{...[ml/ki(n— W)!]a'p"*...} forSym*V* andex- panding out(xa+yf)"weseethatincoordinates thismap maybegiven as Cay) Be ty eh eh From thedefinition, theaction ofPGL,C onP*preserves C,;conversely, since anyautomorphism ofP*fixing C,pointwise istheidentity, from Fact 11.15 itfollows thatthegroup Gofautomorphisms ofP*thatpreserve C,i precisely PGL3€. (Note that conversely ifWisany (n+ 1)-dimensional representation ofSL,€ and PW=P*containsarationalnormalcurveof degreempreservedbytheactionofPGLC,thenwemusthaveW=Sym"V; weleave thisasanexercise*) Whenn=2,Cistheplaneconicdefinedbytheequation 2NotethatanyconfusionbetweenPHandPW’iscelativelyharmlesforushere,sincetherepresentations Sym aeomonpic tothee dua $113.ALittleGeometsicPlethysm Iss F(Zo,24,23)=ZoZa~Z}=0. Forn=3,CisthetwistedcubiccurveinP?,andisdefinedbythreequadraticpolynomials ZZ1—Z}, Zely-ZZy and ZZ, ~Z3. More generally, therational normal curve isthecommon zero locus ofthe 2x 2minors ofthe matrix Zales -Zyas that i,thelocus where therank ofMis1 Back toPlethysm Westart with Example (11.12). Wecaninterpret thedecomposition giventhere(orratherthedecomposition ofthe representation ofthecorresponding Liegroup SLC) geometrically viatheVeronese embedding 13:P'+P?.Asnoted,SLCactsonP?=P(Sym?V*) asthegroupofmotionsofP?carrying theconic curve C,into itself. Itsaction onthespace Sym?(Sym?V)) of quadratic polynomials onP?thus must preserve theone-dimensional sub-spaceCFspannedbythepolynomial FabovethatdefinestheconicC;.Atthesame time, weseethat pullback via1defines amap from thespace of‘quadraticpolynomials onPtothespaceofquarticpolynomials onP*,whichthaskernel €F;thus, wehave anexact sequence 0+€=Sym°V -+Sym?(Sym*V)) ++Sym*¥ +0, Which implies thedecomposition ofSym?(Sym?V)) described above. ‘Note that what comes tousatfirstglance isnotactually thedirect sumdecomposition (11.12)ofSym?(Sym?V)}, butjusttheexactsequenceabove.‘Thesplitting ofthissequence ofSL,C-modules, guaranteed bythegeneral theory, islessobvious. Forexample, wearesaying thatgiven aconic curve C intheplane P2,there isasubspace Ucofthespace ofallconics inP?, complementary totheone-dimensional subspace spanned byCitself and invariant under theaction ofthegroup ofmotions oftheplane P?carrying Cinto itself. Isthereageometricdescriptionofthisspace?Yes:thefollowing Proposition gives one. Proposition 11.16. Thesubrepresentation Sym*V Sym?(Sym?)is thespace conics spanned bythefamily ofdouble lines tangent totheconic C=Cy. Proor. One way toprove thisistosimply write outthissubspace incoor- dinates: interms ofhomogeneous coordinates Z,onP?asabove, thetangentlineotheconicCatthepoint[1,,#*]istheline 136 1,Representations ofhye Ly={2:02 ~202, + Z,=0). The double line21,is,thus, theconic with equation a°Z} —40°Z)Z, +20°ZpZ, +Aa?Z} —aZ,Z, +ZF=O. ‘Thesubspace these conics generate isthusspanned by23,ZZ, ZZa. Zi, andZpZ, +22}. Byconstruction, thisisinvariant under theaction ofSLaC, and itisvisibly complementary tothetrivial subrepresentation C-F = (ZZ ~Z3). Forthose familiar with some algebraic geometry, itmay notbenecessarytowriteallthisdownincoordinates: wecouldjustobservethatthemapfromtheconic curve Co theprojective space P(Sym*(Sym?V)) ofconics inP? sending each point p€Ctothesquare ofthetangent linetoCatpisthe restriction toCofthequadratic Veronese map P?-»P®,andsohasimage aquarticrationalnormalcurve.Thisspansafour-dimensional projective sub-space ofP(Sym*(Sym?V)}, which must correspond toasubrepresentation isomorphic toSym*¥. o We will return tothis notion inExercise 11.26 below. ‘Wecan,inasimilarway,describethedecomposition ofall thesymmetric powers oftherepresentation W=Sym?V; inthegeneral setting, thegeo- ‘metric interpretation becomes quite handy. Forexample, wehave seen that thethird symmetric power decomposes ‘Sym?(Sym?V) =SymV@Sym*¥. This isimmediate from thegeometric description: thespace ofeubics inthe planeP?naturallydecomposes intothespaceofcubicsvanishingontheconic C=Cy,plusacomplementary spaceisomorphic(viathepullbackmap#$)o thespaceofsexticpolynomials onP;moreover,sinceacubicvanishingon €,factors into thequadratic polynomial Fandalinear factor, thespace of cubics vanishing ontheconic curve CcP*maybeidentified with thespace oflines inP2 ‘One more special case: from thegeneral formula (L1.14), wehave ‘Sym*(Sym?V) =Sym*V @Sym*V@Sym°V. Again, this iseasy toseefrom thegeometric picture: thespace ofquartic polynoinials onP?consistsoftheone-dimensional spaceofquartics spannedbythesquareofthedefiningequationFofCitself,plusthespaceofquartics vanishing onCmodulo multiples ofF2,plusthespace ofquattics modulo those vanishing onC.(We usetheword “plus,” suggesting aditect sum, but asbefore only anexact sequence isapparent). Exercise 11.17. Show that, ingeneral, theorder ofvanishing onCdefines a filtration onthespace ofpolynomials ofdegree ninP%,whose successivequotientsarethedirectsumfactorsontherighthandsideofthedecompos-tion ofExercise 11.14 S113. ALittle Geometric Plethysin 137 Wecansimilarly analyze symmetric powers oftherepresentation U= ‘Sym?V.Forexample, since Uhaseigenvalues —3,—1,1.and 3,thesyminetricsquareofUhaseigenvalues —6,—4,~2(twice),O(twice),2(twice),4,and6;diagrammatically, wehave SPE OPO Oboes ‘This implies that Sym?(Sym?¥) 2Syn°V@Sym?¥. (118) ‘Wecaninterpret thisinterms ofthetwisted cubic C=Cc P*asfollows:thespaceofquadraticpolynomials onP?contains, asasubrepresentation, thethree-dimensional vectorspaceofquadricscontaining thecurveCitself;andthequotient isisomorphic, viathepullback map 1$,tothespace ofsextic polynomials onP*, Exercise 11.19*. Bytheabove, theaction ofSLC onthespace ofquadric surfaces containing thetwisted cubic curve Cisthesame asitsaction on P(Sym?V*)=P.Makethisexplicitbyassociatingtoeveryquadriccon- taining Capolynomial ofdegree 2onP',uptoscalars. Exercise 11.20°, The direct sum decomposition (11.18) says that there isalinearspaceofquadricsurfacesinP?preservedundertheactionofSLCandcomplementary tothespace ofquadrics containing C.Describe this space. Exercise 11.21. Theprojection map from Sym?(Sym?¥) toSym?V given by thedecomposition (11.18) above may beviewed asaquadratic map from thevectorspaceSym?tothevectorspaceSym?V.Showthatitmaybegiveninthese terms astheHessian, that is,byassociating toahomogeneous cubicpolynomial intwovariablesthedeterminant ofthe 2x2matrix ofits second partials. Exercise11.22.Themapintheprecedingexercisemaybevievtedasassociatingoanunordered triple ofpoints (p,q.r} inP'anunordered pairofpoints {5,1} PY.Show thatthispairofpoints isthepairoffixed points ofthe automorphism ofPpermuting thethree points p,4,andrcyclically Exercise 11.23*. Show that Symm?(Sym*V) =Sym*V@Sym5V@Sym?V, andinterpret this interms ofthegeometry ofthetwisted cubic curve, Inparticular, showthatthespaceofcubicsurfacescontaining thecurveisthedirectsumofthelasttwofactors,andidentifythesubspaceofcubiescorre-sponding tothelastfactor. 138 11,Representations ofsl,€ Exercise 11.24. Analyze therepresentation Sym*(Sym*V) similarly. Inpar- ticular, show that itcontains atrivial one-dimensional subrepresentation. ‘The trivial subrepresentation ofSym*(Sym?¥) found inthelastexercise hhasaninteresting interpretation. TosaythatSym*(Sym?V) hassuch an invariant one-dimensional subspaces tosaythatthere existsaquarticsurface inP?preserved under allmotions ofP?carrying therational normal curve C= Cyintoitself. What isthissurface? Theanswer issimple: itisthe¢angent developable tothetwisted cubic, that is,thesurface given astheunion ofthe tangent lines toC. Exercise11.25*.Showthattherepresentation Sym?(Sym*V) containsatrivialsubrepresentation, andinterpret thisgeometrically. Problem11.26.Anotherwayofinterpreting thedirectsumdecomposition ofSym?(Sym?V) geometrically istosaythat given aconic curve C=P?and given four points onC,wecan find aconic C’=C(C;py,..-5Ps)<P intersecting Cinexactly these points, inaway thatispreserved bytheactionofthegroupPGL5€ofallmotionsofP?(i.foranymotionA:P?-+P?oftheplane,wehaveA(C'(C;py,»Pa))=C(AC;APs.»APa))-Whatisadescription ofthis process? Inparticular, show that thecross-ratio ofthe fourpointsp,onthecurveC”mustbeafunctionofthecross-ratio ofthep,onC,and find thisfunction. Observe also that thisprocess gives anendomorphism ofthe pencil {COPY py PeeCh=PE ofconics passing through anyfourpoints p,¢P?,What isthedegree ofthis ‘endomorphism? ‘The above questions have alldeat with thesymmetric powers ofSym*¥. ‘There are also interesting questions about theexterior powers ofSym*¥: Tostart with, considertheexteriorsquareA?(Sym?V).Theeigenvaluesofthis tepresentation arejustthepairwisesumsofdistinctelementsof(3,1,—1,~3}. thatis,4,2,0(twice), ~2,and—4;wededucethatAXSym?)=Sym*V@Sym?V, (1127) Observe inparticular thataccording tothisthere isaskew-symmetric bilinearformonthespaceU=Sym?Vpreserved(uptoscalars)bytheactionofSLC.What isthis form? One way ofdescribing itwould beinterms ofthetwisted ccubie: themap from Ctothedual projective space (P°)* sending each point eC totheosculating plane toCatpextends toaskew-symmetric linear isomorphism ofP?with (P2)*. Exercise 11.28. Show that alineinP?isisotropic forthisform ifandonly if,viewedasanelementofP(\*U),itliesin thelinearspanofthelocusoftangent lines tothe twisted cubic. §113. ALittle Geomettic Pethysm 159 Exercise 11.29, Show that theprojection onthefirst factor inthedecomposi- ‘ion (11.27) isgiven explicitly bythemap FAGreF-dG—G-dF andsayprecisely what thismeans. Exercise 11.30, Show that,ingeneral,therepresentationA?(Sym"V)hasasa directsumfactor therepresentation Sym?*~*V, andthatthe projection onthis factor isgiven asinthepreceding exercise. Find theremaining factors of 12(Sym"V), andinterpret them. More onRational Normal Curves Exercise 11.31. Analyze ingeneral therepresentations Sym?(Sym"V}; show, using eigenvalues, that wehave Sym™(Sym"V) = Sym*"~*¥, Exercise 11.32%. Interpret thespace Sym?(Sym"V) ofthepreceding exerciseasthespaceofquadrisintheprojectivespaceP*,andusethegeometryoftherationalnormalcurveC=C,<P*tointerpretthedecomposition ofthis representation into irreducible factors. Inparticular, show that direct sum @sme isthe spaceofquadraticpolynomials vanishingontherationalnormalcurve; and that the direct sum @symrey isthespaceofquadtiescontaining thetangentialdevelopable oftherationalnormal curve, that is,theunion ofthe tangent fines toC.Can you interpretthesumsfor«>kfork>27 Exercise 11.33, Note that byExercise 11.11, thetensor power ‘Sym'V @Sym" ahways contains acopy ofthetrivial representation; and that byExercises 11.30 and11.31, thistrivial subrepresentation willieinSym"(Sym"V) ifnis cexenandin\2Sym*V) ifisodd. Show thatinether case, thebilinear formonSymn"VpreservedbySLCmaybedescribed astheisomorphism ofP*with(PP carrying each point poftherational normal curve C< P*into the ‘oscuating hyperplane toCatp. Comparing Exercises 11.14 and 11.31, weseethat Sym?(Sym"v) = Sym"(Sym*V}; apparently coincidentally. This isin fact«special case ofa ‘more general theorem (efExercise 6.18) 160 1,Representations of#y€ Exercise 11.34. (Hermite Reciprocity). Usetheeigenvalues ofHtoprove the isomorphism Symm'(SymrV) =Sym"(Sym*V). Canyouexhibitexplicitlyamapbetweenthesetwo? Note that intheexamples ofHermite reciprocity wehave seen, it seems completely coincidental: forexample, thefactthat therepresentations Sym?(Sym*V) andSym*(Sym?V) both contain atrivial representation cor- responds tothefacts that thetangential developable ofthetwisted cubic in P?hasdegree 4,while thechordal variety oftherational normal quartic in P*hasdegree 3. Exercise 11.35*. Show that A"(Sym"V) =Sym™(Sym""?-"V),, WewillseinLecture23thatthereisauniqueclosedorbitinP(WV)forany irreducible representation W.Fornow, wecandothefollowing special case Exercise 11.36. Show that theunique closed orbit oftheaction ofSLC on theprojectivization ofanyirreducible representation isisomorphic toP' (these aretherational normal curves introduced above). LECTURE 12 Representations ofsI,C, Part I “Thislecturedevelopsesullsforsl,€analogoustothoteof1.1(thoughnotinexactly thesame orden. This involves generalizing some ofthe basic terms ofS (eg, theottoofeigenvalueandeigenvectorhavetoberedefined),buthebasicideasarein samesensealeadyin11,Certainlynotechniquesareinvolvedbeyondthoseof1.1. Wecomenowtoasecondimportantstageinthedevelopment ofthetheory: inthefollowing, wewilltakeouranalysisoftherepresentations ofs1,Cand‘sechowitgoesoverinthenextcase,thealgebra sl,C.Aswewillsee,anumber ofthebasic constructions need tobemodified, oratleast rethought. There are,however, twopieces ofgood news that should beborne inmind. First, wewillarrive, bytheendofthefollowing lecture, ataclassification ofthe Fepresentations ofs1,C that isevery bitasdetailed and explicit astheclassifi- cation wearrived atpreviously forsl,C. Second, once wehave redone out analysis inthiscontext, wewillneed tointroduce nofurther concepts tocarry outtheclassification ofthefinite-dimensional representations ofallremaining semisimple Liealgebras. ‘Wewillproceed byanalogy with theprevious lecture. Tobegin with, we started outouranalysis ofs1,C with thebasis (H,X,Y}fortheLiealgebra; wethen proceeded todecompose anarbitrary representation Vofsl,€ into 4direct sum ofeigenspaces fortheaction ofH.What element ofs1,C in particular willplay therole ofH?The answer—and thisisthefirst and perhaps most wrenching change from theprevious case-—is that noone clement really allows ustoseewhat isgoing on.' Instead, wehave toreplace "Thin inert true awewilsefromthelowingannyiranydngonlmati seen arindependent over, thon theation of omany eprcenation Vo a Aecrmivs therepresentation (eiweknowtheeigenalesefweknow7)Ba(wewl tteyingtoearrylouinractcewouldbesheeperersty 162 12,Representations ofsf,C, Part thesingle element H€sl,C with asubspace hcslyC, namely, thetwo- dimensional subspace ofalldiagonal matrices. The idea isabasic one; it comes down totheobservation that commuting diagonalizable matrices are simultaneously diagonalizable. This translates inthepresent circumstances tothestatement thatanyfinite-dimensional representation Vofsly€admitsadecomposition V=@V,,whereeveryvectorv€V,isaneigenvector foreveryclement Heb, ALthispoint some terminology isclearly inorder, since wewillbedealing, with the action not ofasinglematrixHbutratheravectorspace}ofthem. Tobegin with, byaneigenvector forhwewill mean, reasonably enough, a vector v€Vthat isaneigenvector forevery H€bh.Forsuch avector owecan write H(o) =aH)», (2 where a(H) isascalar depending linearly onH,ic.,«€h*.This leads toour second notion: byaneigenvalue fortheaction ofhwewillmean anelement @€b® such that there exists anonzero element v€Vsatisfying (12.1), and by theeigenspace associated totheeigenvalue «wewillmeanthesubspace ofallvectors v€Vsatisfying (12.1) Thus wemay phrase thestatement above as (12.2) Anyfinite-dimensional representation Vofsl;C hasadecomposition v= OM where V,isan eigenspace for andarangesoverafinitesubsetofi. ‘This is,infact,aspecialcaseofamoregeneralstatement: foranysemisimple Liealgebra g,wewillbeable tofind anabelian subalgebra h<g,such that theaction ofhonany g-module Vwillbediagonalizable, ic.,wewill have a directsumdecomposition ofVintoeigenspaces ¥,forb.Having decided what theanalogue forsl,C ofH€sl,€ is,letusnow consider what willplay therole ofXand ¥.The keyhere istolook atthe commutation relations (H,X)=2X and (H,¥}=~—2Y inst,C. The correct way tointerpret these isassaying that Xand Yare eigenvectors fortheadjointactionofHonsi,C.Inourpresentcircumstances, then, wewant tolook foreigenvectors(inthenewsense)fortheadjointaction ofhonsls€.Inotherwords,weapply(12.2)totheadjoint representation of s1y€ toobtain adecomposition s,€ [email protected] (123) where«rangesoverafinitesubsetofh*andbactsoneachspaceg,byscalarmultiplication, ie,forany Hehand Yeg,, [H, ¥]=ad(Hy(¥)=a(t)¥. Thisisprobablyeasiertocarryoutinpracticethanitistosay;wearebeing 12.Representations ofs,€, Part 163, longwinded here because once thisprocess isunderstood itwill bestraight- forward toapply ittotheother Liealgebras. Inanycase, todo itinthepresentcircumstances, wejustobservethatmultiplication ofamattixMontheleft bbyadiagonal mattix Dwith entries a,multiplies theithrowofMbya,while ‘multiplication ontheright multiplies theithcolumn bya;ithe entries ofM arem,;,theentriesofthecommutator (D,M]arethus(a;~a,)m,,.Weseethen that thecommutator [D,M]willbeamultiple ofMforallDifand onlyifallbutoneentryofMarezero.Thus,ifweletE,bethe3x3matrixwhose (idthentryis{andallofwhoseotherentriesare0,weseethattheE,exactly generatetheeigenspaces fortheadjointactionofbong.Explicitly, wehave a, 0 0) b=4[0 a, 0):a,40,40,=0 0 0 a and sowe can write he=CfLy,Ly,La}Ly+La+Ly=0)}, where a, 0 0 LJ0a, 0}=a, 0 0a ‘The linear functionals« h*appeatinginthedirectsumdecomposition (12.3) arethusthesixfunctionals L,—L;thespace qy,.., willbegenerated bythe element E,,,.Todraw apicture tasks .ARK.a(124) ‘The virtue ofthisdecomposition andthecorresponding picture isthat wecanreadofffromitprettymuchtheentirestructureoftheLiealgebra.OF 164 12,Representations ofsy€,PartI course, theaction ofhon9isclear from thepicture: bcarries each ofthe subspaces g,intoitself, acting oneach g,byscalar multiplication bythelinear functional represented bythecorresponding dot.Beyond that,though, wecanalsosee,muchasinthecaseofrepresentations ofs1,C,howtherestoftheLiealgebra acts. Basically, weletXbeany element ofg,and askwhere ad(X) sends agiven vector Y€gp:theanswer asbefore comes from knowing howby actsonad(X)(V). Explicitly, weletHbeanarbitrary element ofhandason page 148 wemake the Fundamental Calculation (second time): (A,OX,3)=X,CH,0)+CCH,XD.7 =LX,UH) V+ Cath)X,1 =(o(H) +ACH))-LX, ¥}. Inother words, [X, Y]=ad(X)(¥) ésagain aneigenvector forb,with eigen- value a+f.Thus, Ad(9.): 86>Ber45 inparticular, theaction ofad(g,) preserves thedecomposition (12.3) inthe sense thatitcarries each eigenspace g,intoanother. Wecaninterpret thisintermsofthe diagram (12.4) ofeigenspaces bysaying thateach g,acts, 0to speak, by“translation”; thatis,itcarries each space q,corresponding toadotinthediagramintothesubspaceg,,,corresponding (othatdottranslated by«.Forexample, theaction ofqz,-, maybepictured as ° tasksa 7 a‘ boty Ou byob ets tek 12.Representations ofeC,PartI 165 iceitcarries1-1,iM€0614-145914-14IMOB;HIMOB1,-14015-14M0Br,-Lwandkills1,15: 91y-14»8Nd 8,1,Ofcourse,notallthedatacanbereadoff ofthediagram, atleast onthebasis onwhat wehave said sofar.Forexample,wwedonotatpresentseefromthediagram thekernelofad(g,,-.,) on},though‘wewill see later how toread this offaswell.Wedo,however,haveatleasta prettygoodideaofwhoisdoingwhattowhom.Pretty much thesame picture applies toany representation Vofl,C: we start from theeigenspace decomposition V=@Y, fortheaction ofthatwe sawin(12.2). Next, thecommutation relations forsly tellusexactly how the remaining summands ofthedecomposition(12.3)ofsl,€actonthespaceV, ‘andagain wewillseethateach ofthespaces g,actsbycarryingoneeigenspace ¥,intoanother. Asusual, foranyX€g,andv€Vjwecantellwhere XwillsendvifweknowhowanarbitraryelementH«hwillactonX(o).Thiswecandetermine bymaking the Fundamental Cateulation (third time}: H(X(0) =X(H(0) +LH,XI) =X(BUH): 0)+(aC) XY(0) =(@(H) +ALI): X(0 Weseefrom thisthat X(0) isagain aneigenvector fortheaction ofbywitheigenvalue a+fiinotherwords,theactionofg,cartiesVst0V,,4.Wecanthusrepresenttheeigenspaces V,ofVbydotsinaplanediagramsothateach{acts again “bytranslation,” aswedidforrepresentations ofs1,C inthe preceding lecture and theadjoint representation ofs1,C above. Just asin the case ofsls (page 148), wehave Observation 12.6. The eigenvaluesaoccurringinamirreductblerepresentation ofs1yC differ from oneother byintegral linear combinations ofthevectors L,-Leb [Note thatthese vectors L—Lygeneratealatticeinh*,whichwewilldenote byAp,and that allthe lieinsome translate ofthislattice. ALthispoint, weshould begin tointroduce some oftheterminology thatappearsinthissubject.Thebasicobjecthere,theeigenvalue «€h*oftheactionofhonarepresentation Vofg,iscalledaweightoftherepresentation;thecorresponding eigenvectors in¥,arecalled, naturally enough, weight vectors andthespaces ¥,themselves weight spaces. Clearly, theweights that ‘occur intheadjoint representation arespecial; these arecalled theroots of theLiealgebra and thecorresponding subspaces g,¢groot spaces; by 166 12,Representations ofsl,PartI convention, zero isnotaroot. The lattice Ay<b*generated bytheroots « iscalled the root lattice. “Toseewhat thenext stepshould be,wegoback totheanalysis ofrepresen- {ations ofsl,C.There,atthisstagewecontinuedouranalysisbygoingtoan extremal eigenspace V,andtaking avector v€V,.Thepoint wasthatsince V, wasextremal, theoperator X,which would carry V,toV2. would have to Kill;s0 that »would bethen both aneigenvector forHandinthekernelof X.Wethen saw that these two facts allowed us(ocompletely describe the representation Vinterms ofimages ofv. What would betheappropriately analogous setup inthecaseofst,C? To start atthebeginning, there isthe question ofwhat wemean byextremal: in thecaseofst3C, since weknew thatalltheeigenvalues were scalars differing, byintegral multiples of2,there wasnotmuch ambiguity about whatwemeantbythis.Inthepresentcircumstance thisdoesinvolveaprioriachoice(though asweshall seethechoice does notaffect theoutcome). wehave tochoose a direction, and look forthefarthest ainthat direction appearing inthe decomposition (12.3). What thismeans isthat weshoukd choose alinear functional [Ag R, extend itbylinearity toalinear functional [:*—+C, and then forany representation ¥weshould gotothecigenspace F,forwhich therealpart of Ifa)ismaximal? Ofcourse, toavoid ambiguity weshould choose Itobe irrational with respect tothelattice A,that is,tohave nokernel. What isthepoint ofthis? The answer isthat, just asinthecase of a. representation VofslC wefound inthis way avector v¢¥that was simultaneously inthekernel oftheoperator Xandaneigenvector for#1,ip thepresent casewhat wewillfindisavector v€¥,thatisancigenvector for 4,andatthesame timeinthekernel oftheaction ofagforevery fhsuchthat[(f)>O—thatis,chatiskilledbyhalftherootspaces94(specifically, therootspacescorresponding todotsinthediagram(12.4)lyingin@halfplane).Thiswilllikewise giveusanearly complete description ofthe representation V. ‘Tocarry thisoutexplicitly, choose ourfunctional Itobegiven by HayLy +agLy +ayLs)=aay+bay+cay, wherea+b+¢=0anda>b>,s0thatthespacesg,<gforwhichwe hhave(a)>0arethen exactly 1,15. @cz-r4» ANUGy,-14; they correspond to ‘matrices with onenonzero entry above thediagonal. 2Thereaversuscomplexbusinesiarehertngsince(twillturnoutveryshortyallhesigervlues aactualy occutring inanyrepresentation wilinfact bei theeaGinfc, the rationalnearspanof 12Representations ofss€,PartE 167 Lys HOOK ™ ~-V—-\- Ino Thus, fori<j,thematricesE,,,generatethepositiverootspaces,andtheE,, ‘generate thenegative root spaces. We'sct Hy=UB Bud =Bia—Bx (128) Now let¥beany irreducible, fnite-dimensional representation ofs1,C. ‘The upshot ofalltheabove isthe Lemma 12.9. There isavector veVwith theproperties that (0visaneigenvectorforb,te.v€¥,forsomea;and i)viskilled byEy,3,Es,andEy.» Foranyrepresentation VofslyC, avector veVwith these properties is called ahighest weight vector.Inthecaseofs1,C,havingfoundaneigenvector vforHkilledbyX,wearguedthattheimagesof»undersuccessive applications of¥generated therepresentation. The situation here isthesame: analogous toClaim 11.4 we have (Claim12.40.LetVbeanirreducible representation ofslyC,andveVahighestweight vector. Then Visgenerated bytheimages ofvunder suecessive applica- tionsofthethreeoperatorsE,.,Ey,,,andEs,,. Before wecheck theclaim, wenote three immediate consequences. First, itsays that alltheeigenvalues fhe*occurring inVlieinasortof{-planewithcornerata 168 12,Representations ofl,C,PartI —LED Second, weseethat thedimension of¥,itself is1,90that oistheunique eigenvector with thiseigenvalue (uptoscalar, ofcourse). (We willseebelow that infactoistheunique highest weight vector ofVuptoscalars; sce Proposition 12.11) Lastly itsays thatthespaces Vesyty-14 800Veunty-t) areallatmost onedimensional, since theymust bespanned by(E,,)"(0) and (Es,,F(0, respectively. PROOF OFCLAIM 12.10. This isformally thesame astheproof ofthecorre-spondingstatement forslyC: weargue that thesubspace WofVspanned by images ofvunder thesubalgebra ofs1,€ generated byE,, Es,1, andEy,infac,preservedbyallofs1yCandhencemustbeallofV.'Todothiswejust hhave tocheck thatEy», E.s, andBy,ycarryWintoitself(infactitisenough todothis forthefirst two, thethird being their commutator), and this is straightforward. Tobegin with, vitself isinthekernel ofE,»,Es», ndEy», sothereisnoproblemthere,NextwecheckthatE,,(0)iskeptin W:wehave Ex,2(E2,1(0)) =(Ex,(Es,2(0))+CEe.2»Ea,100) =(Es2Ea,D0sinceE,,(0)=0and[Ey,5,By,.J€b:and Ey,s(E2,1(0)) =(Ex, (Es, s(0))+(Ea,9»E2100) =0 sinceE,s(t)=0and[E3,s,2,1]=0.Asimilarcomputation showsthatE,,0) ialsocarried into VbyEy, andE,,», More generally, wemay argue theclaim byasort ofinduction: welet denote anyword oflength morlessintheletters E,,andEs,andtakeW, tobethevector space spanned bythevectors w,(0) forallsuch words; nolethatWistheunionofthespacesW,,sinceE,,;isthecommutator ofE,»andEq,,. Weclaim that Ey, and Es,5 carry W,into W,... Toscethis, wecan 12,Representations ofsy€, PartI 16 write w,aseither E>, ©¥y-1 OFEs, 0m,-,3 ineither case w,_,(s) willbean cigenvector forbwith eigenvalue fforsomefntheformercasewehave Elo) =By22.10%) =Ea,s(E4.20%-())+CErarBa,1M0m-1(0)) ©Ea,(Wy-2)+BUTE1,2Ea,11)M10) an since (E,25 B,. €and Ez,s(waO))=Ea.3(E2,10e-())) =Ea,1(Ea,3044-10) +CEr,s»Ea,110%4-100)) ©Exi(W-2) eM since [E3,s, E,,,] =0.Essentially thesame calculation covers thelatter caseWy=Ey,3©tye,establishing theclaim. o Thisargumentshowsalittlemore;infact,itproves Proposition 12.11. IfVisanyrepresentation ofslyC andv€Visahighest weight vector, then thesubrepresentation WofVgenerated bytheimages ofvbysuccessive applications ofthethree operators E,y, Es,., andEy, is itreducible. Proor.Letabetheweightofv.TheaboveshowsthatWisasubrepresenta-tion, and itisclear that W,isone dimensional. IfWwere notirreducible, wewouldhaveW=W"@W"forsomerepresentations W’andW".ButsinceProjection toW’and W"commute with theaction ofb,wehave MW,= W;®Wz.Thisshowsthatoneofthesespacesiszero,whichimpliesthatvbelongs toW"orW",andhence that Wis W”orW". o Asacorollary ofhispropositionweseethatanyirreduciblerepresentation ofs€ hasaunique highest weight vector, uptoscalars; more generally, the setofhighest weight vectors inVforms aunion oflinear subspaces “Py corresponding totheirreducible subrepresentations Wof¥,with thedimen sion ofyy equal tothenumber oftimes Wappears inthedirect sum decomposition ofVintoirreducibles. ‘What dowedonext? Well, letuscontinue tolook attheborder vectors (E,,:)"(0). Wecallthese border vectors because theylivein(and, aswesaw, span)acollectionofeigenspaces a,Qes1-14»Ses2yiy-tyy~~thatcorrespond topoints ontheboundary ofthediagram above ofpossible eigenvalues of¥,Wealso know that they span anuninterrupted string ofnonzero cigen-SPOSBysauy-14) %CK=O,Lynyuntilweget{0thefirstmsuchthat 170 12,Representations ofsl,C, PattI (E2.,)"(0) =0,afterthatwehave Gysai..-t,) =(0)forallk>m.Thepicture isthus: RRQOOO wherewehavenodotsabove/totherightofthe bold line, and nodots onthat line other than the ones marked. Theobvious question now ishow Jong thestring ofdots along thislineis. ‘One way toanswer thiswould betomake acalculation analogous totheoneintheprecedinglecture:usethecomputation madeabovetosayexplicitlyforanykwhatmultipleof(E,,)*~(0) theimageofE,,.}(o) under themapEy,» is,and usethefact that (Ea, )*(e) =0todetermine m.Itwill besimpler— and more useful ingeneral—if instead wejust use what wehave alreadyJearnedaboutrepresentations ofsis€.Thepointis,theelements Ey.ondEa,1.together withtheir commutator (Ey,2,Ey,.}=Hy3»panasubalgebraofslyC isomorphic tosl,€ viaanisomorphism carrying Ey, By,, andHy,» tothe elements X,¥andH.Wewilldenote thissubalgebra by6. (thenotationmayappearawkward, butthisisaspecialcaseofageneralconstruction). Bythedescription wehave already given ofthe action ofs1yC ontherepresenta- tionVinterms ofthedecomposition V=@)¥,, wesecthatthesubalgebra51,1,Willshiftcigenspaces V,onlyinthedirectionofLy—L,;inparticular,thedirectsumoftheeigenspaces inquestion,namelythesubspace W=@oeonts-u9 (1213 of¥willbepreserved bytheaction ofs,,-,,- Inother words, Wis a representation of8,1, =s1,€ andwemaydodice from thisthattheeigen: ealues ofH,, onWare integral, andsynnmetric with respect tozero. Leavingasidetheintegrality forthemoment,thissaysthatthestringofdotsindiagram(12.12) must besymmetricwithrespecttotheline<H,,.,L)=Ointheplane by*.Happily (though bynomeans coincidentally, asweshall see), thislineisperpendicular tothelinespannedbyLy—Lyinthepicturewehavedraw,80wecan saysimply that thesiring ofdots occurring indiagram (12.12) ispreserved underreflectionintheline(Hh,2,L}=0.Ingeneral, foranyi#/ theelemenis E,,andF;,,together with their commutator [Ej 5, =Hi.span asubalgebra 9,1, ofsly isomorphic tosl;Cviaanisomorphism carryingE;»E,,.,andH,,totheelementsX,¥,and H,(Note that 1,,= —H,,.) Analyzing theaction ofthesubalgebra <H,_3, L)=0inb*.Thepicture isthus SODKSOYOARVAVAVAVAV VIED ATAYANAKA Letusnowtakealook atthelasteigenspace inthefirststring, thatis,Vy where ntisasbefore thesmallest integer such that (E,,,)"(o) =0and f=a+(m—I)(Ly—Ly).Ifv'€Vyisanyvector,then,bydefinition, wehave thebold lineindiagram (12.12), wehaveaswellthatEys(0')=By,s(0")=0. ‘Thus, v’,like vitself, satisfies thestatement ofLemma 12.9, except forthe exchange oftheindices2and1;orinotherwords,ifwehadchosenthelinear functional |abovedifferently —precisely, withcoefficients b>a>c—then Tespect tothevector v'instead ofv,wewould findthtalleigenvalues ofV ‘occur below ortotheright ofthelines through finthedirections ofL,—Ly were symmetric about thelines (H,,,L) =0and (Hy,L)=0,respec tively. The picture now is POSOLOseein VAVAVAVAVAY/ WEE seared attheendofthestringofeigenvalues {fi+k(Ls—L,)}wewillarriveata eigenvalues inVwillbebounded byahexagon symmetric withrespecttothe lines (H,,, L)=0andwithonevertex ata;indeed, thischaracterizes the ofisometries ofthe plane generated byreflections inthese three lines. VAALKOXLDLOO QADY crete iROEWIDELYiyVETANAYA Wewillseeinamoment thatthesetofeigenvalues willinclude allthepoints multiplicity one. ‘oftheelements H1,,,mustbeintegers;itisnothardtoseethatthismeansthatalltheeigenvalues occurring in(12.2) must beintegral linear combinations of theL,,i¢.,interms ofthediagrams above, alldots must lieinthelattice Aw‘ofinterstices (asindeed wehave been drawing them). Thus, wehave Proposition 12.15. Alltheeigenvalues ofany irreducible finite-dimensional Thisisexactlyanalogous tohesituation oftheprevious lecture:therewesaw that theeigenvalues of1inanyirreducible, finite-dimensional representa- andwerecongruent tooneanother modulo thesublattice Ay=2:Zgenerated 12,Representations ofst,©, Part I 1% bytheeigenvalues of/under theadjoint representation, Note that intheeaseofsl,CwehaveAy/Ag2Z/2,whileinthepresentcasewehaveAy/Ay2Z/3 wewillscelater how thisreflects ageneral pattern. The lattice Aviscalled theweight lattice. Exercise 12.16, Show thatthetwoconditions thattheeigenvalues ofVare congruent toone another modulo Agand arepreserved under reflection in thethree lines Hy, L)=0imply thattheyalllieinAw,andthat, infact, thischaracterizes Ay. Tocontinue, wecangointo theinterior ofthediagram (12.14) ofeigen- values ofVbyobserving that thedirect sums (12.13) arenottheonly visible subspaces ofVpreserved under theaction ofthesubalgebras s,,_.,; more generally, foranyf¢b*appearinginthedecomposition (12.2)andanyi,jthe direct sum Wo@oonietp willbearepresentation of¢,-1,(notnecessarily irreducible, ofcourse);inparticular itfollows thatthevalues ofkforwhich Vysau,-19 %(0)form an iunbroken string ofintegers. Observing that iffisany ofthe“extremal” cigenvalues pictured indiagram (12.[4), then thisstring willinclude another; 0thatalleigenvalues congruent tothedots pictured indiagram (12.14) and lying intheir convex hull must also occur. Thus, thecomplete diagram of eigenvalues willlook like Wecan summarize thisdescription in Proposition 12.18, LetVbeanyirreducible, finite-dimensional representationGSC. Then forsome a€Ny&*, thesetofeigenvalues occurring inVis 1m 12.Representations ofsi, Part ‘exactly thesetoflinear functionals congruent toamodulo thelattice Ayand Lying inthehexagon with vertices theimages ofaunder thegroup generated by reflections inthelines (Hy, L)=0. Remark,Wedid,intheanalysisthusfar,makeoneapparently arbitrarychoicewhenwedefinedthenotionof“extremal” eigenvalue bychoosingalincarfunctional {on b¢.Weremark here that, infact, thechoice was notasbroad asmight atfirsthave appeared. Indeed, given thefactthat theconfiguration ofeigenvalues occurring inanyirreducible fnite-dimensional representation ofsiyC isalways either atriangle orahexagon, the“extremal” eigenvalue picked outby|willalways turn outtobeoneofthethree orsixvertices ofthisfigure;inotherwords,ifwedefinethelinearfunctional {totakea,L+azLy +ayly 0aa,+bay+cas, then only theordering ofthethree real numbers a,6,andcmatters. Indeed, inhindsight thischoice wascompletely analogous tothechoice wemade (implicitly) inthecase ofs1,C inchoosing oneofthetwodirections along thereal line. Wesaid attheoutset ofthis lecture that ourgoal was toarrive ata description ofrepresentations ofsly ascomplete asthat fors1,€. Wehave now, certainly, ascomplete adescription ofthepossible configurations of cigenvalues; butclearly much more isneeded. Specifically, weshould have anexistence and uniqueness theorem;anexplicitconstruction ofeachrepresentations, analogoustothestatementthateveryrepresentation ofsl,Cisasymmetric powerofthestandard;and{orthepurposeofanalyzingtensorproductsofrepresentations ofsly,weneed adescription notjustofthesetofeigenvalues, butofthemultiplicities with which they occur, (Note that thelastquestion isonethat hasnoanalogue inthecase ofsI,C: inboth cases, anyirreducible representation isgenerated bytaking asingle cigenvector v€¥,and pushing itaround byelements ofg,;butwhereas inthepreviouscasetherewasonlyonewaytogetfromV,{oFy—thatis,byapplyingYoverandoveragain—in thepresentcircumstance therewillbemorethanconeway ofgetting, forexample, rom ¥,t0Yes1,1,3 andthese may yield independent eigenvectors, This hasbeen, however, already toolong alecture,andsowewilldeferthesequestions,alongwithallexamples, tothenext. LECTURE 13 Representations ofsl,C, Part II: Mainly Lots ofExamples Inthislecture wecomplete theanalysis oftheirreducible representations ofs1,C,culminating in§13.2withtheanswerstoallheeofthequestionsraisedattheendofthelasttur:weexplicitlyconstructtheuniquereduciblerepresentation withgiventighest weight andinparticular determine ismultiplicities. Theater (wosectionscorrespondto$11.2and113intheletreonl.Inpartclar,$134,like6113,involves some projective algebraic geometry andmay beskipped bythose towhom this wots. $134: Examples $132: Description ofthe irreducible representations 4133: ATite more plethysm {134 Alitle more geometric pethysm §13.1. Examples ‘Thislecturewillbelargelyconcernedwithstudyingexamples,givingconstruc- tions and analyzing tensor products ofrepresentations ofsl,€. Westart, ‘however, byatleast stating thebasic existence anduniqueness theorem that provides thecontext forthisanalysis Tostate this, recall from theprevious lecture than any irreducible, fnite- dimensional representation ofely hasavector, unique uptoscalars, tha is simultaneously aneigenvector forthesubalgebra hand killed bythethree subspaces G),-149r,-14» ANd9,,,-1,-Wecalledsuchavectorahighestweight vector oftherepresentation V;itsassociated eigenvalue will, ofcourse, be called the highest weight ofV.More generally, inany finite-dimensional representation WofstsC, anyvector 0€Wwith these properties wllbecalledAhighestweightvector;wesawthatitwillgenerateanirreducible sub- 16 13.Representations ofs1,€,PactIt:MainlyLotsofExamples representation VofW.Finally, from thedescription given inthefastlecture ‘ofthepossible configurations ofeigenvalues forarepresentation of#1,C,we soethatanyhighest weight vector must lieinthe({}:plane described bythe inequalities <H,,2,.L) =0and(Hy,s,L) 20, ie,itmust beoftheform (a+ BL, +bLy=aly—bLyforsome pairofnon-negative integers aand 'b.We can now state ‘Theorem 13.1. For any pair ofnatural numbers a,bthere exists aunique Irreducible, finite-dimensional representation Fyofsl, with highest weight aly ~bby. Wewilldefer theproof ofthistheorem until thesecond section ofthis Fecture, notsomuch because itisinanyway difficult butsimply because itis time togettosome examples. Wewill remark, however, that whereas inthe case ofslsC theanalysis that ledtotheconcept ofhighest weight vector immediately gave theuniqueness part oftheanalogous theorem, here toestablishuniqueness wewillbeforcedtoresorttoamoreindirecttrick.Theproofofexistence, bycontrast,willbeverymuchlikethatofthecorresponding, statement forsl,C: wewill construct therepresentations T.,outofthe standard representation bymultilinear algebra. For thetime being, though, wewould liketoapply theanalysis ofthepreviouslecturetosomeoftheobviousrepresentations ofs1,€, partly togain some familiarity with what goes onandpartly inthehopes ofsceing ageneral ‘multiinear-algebraic construction. Webeginwiththestandardrepresentation ofsl,ConV=€°.Ofcourse, theeigenvectors fortheaction off arejust thestandard basis vectors €,,€3, andes;they haveeigenvalues L,La,andL,,respectively. Theweight diagram for Visthus Next,considerthedualrepresentation ¥*.Theeigenvalues ofthedualof arepresentation ofaLicalgebraarejustthenegativesoftheeigenvalues oftheoriginal, sothediagram of¥*is 513.1, Examples ” Alternatively, ofcourse,wecanjustobservethatthedualbasisvectorse?areeigenvectors with eigenvalues —Ly,‘Notethatwhileinthecaseofsl,€theweightsofanyrepresentation weresymmetric about theorigin, and correspondingly each representation was isomorphic toitsdual, thesame isnottruehere (that thediagrams forVand ¥*look thesame isareflection ofthefactthat thetworepresentations are carried into one another byanautomorphism ofsly€, namely, theauto- ‘morphism X++—'X). Observe also that /*isalso isomorphic totherepre- sentation /?Y, whose weights arethepairwise sums ofthedistinct weights of V;andthatlikewise Visisomorphic asrepresentation to\?V*. ‘Next, consider thedegree 2tensor products ofVand *,Since theweights ofthe syimmetricsquareofarepresentation arethepairwisesumsofheweights, othe original, theweight diagram ofSym?¥ willlook like andlikewisethesymmetric squareSym?V*hasweights{—2Ly.Ly~Ly}=(-21, ~21,La}: 178 13,Representations ofs1,€,PartII:MainlyLotsofExamples Weseeimmediately from these diagrams thatSym?¥ andSym?V* are irreducible, since neither collection ofweights istheunion oftwo collectionsarisingfromrepresentations ofs1,€.Asforthetensor product V@ V*, itsweights arejust thesums ofthe weights (L,)ofVwiththose {—L,} ofV*,that is,thefinear functionals L,—L, (each occurring once, with weight vector ¢,@ef)and0(occurring withmulti plicity three, with weight vectors e,@e?).Wecanrepresent these weights by thediagram where thetriple circle isintended toconvey thefactthat theweight space fy isthree dimensional. Bycontrast with thelast twoexamples, this representa- tionisnotirreducible: there isalinear map V@Vee given simply bythecontraction 0@urys do,ut>=ut(o) 813.1, Examples 179 (or,interms oftheidentification V@ V*=Hom(¥, V),bythetrace) that is ‘amap ofsl,C-modules (with €thetrivial representation, ofcourse). The kernel ofthismap isthen thesubspaceofV@¥'*oftracelessmatrices,which isjusttheadjointrepresentation oftheLiealgebrasl,andisirreducible (wecansecthisetherfromourexplicitdescription oftheadjointrepresentation —forexample,Ey,istheuniqueweightvectorfotbkilledbyG1,-14»81,—L5»andg,,-1,—0F ifwetake asknown thefactthatSLC issimple, from the factthatasubrepresentation oftheadjoint representation isanideal inaLie algebra, andexponentiates toanormal subgroup, f.Exercise 8.43.)(Physicists calthisadjointrepresentation ofsls€ (orSU(3)) the“eightfold way,” and relate itsdecomposition tomesons and baryons. The standard representation Visrelated to“quarks” and V*to“antiquarks.” Sec[S-W], [Mack}) (We note that,ingeneral,ifVisanyfaithfulrepresentation ofaLiealgebra, theadjoint representation will appear asasubrepresentation ofthetensor Vevey Letuscontinue now with some ofthetriple tensor products ofVand ¥*, ‘which will bethelastspecific cases welook at.Tobegin with, wehave the symmetric cubes Sym? VandSym? *,with weight diagrams and 180 13,Representations ofssC, Part I:Mainly Lots ofExamples respectively. Ingeneral itis clear that, interms ofthedescription given in thepreceding lectore ofthepossible weight diagrams ofirreducible repre- sentations ofsly, thesymmetric powers ofVand ¥*will beexactly the representations with triangular, asopposed tohexagonal, diagrams. Italso follows from theabove description and thefact that theweights of thesymmetric powers Sym" occur with multiplicity 1that Sym"V and Sym*V* areallirreducible, ie,wehave, inthenotation ofTheorem 13.1, Sym'V =F and Sym*V* =Ty Byway ofnotation, wewilloften write Sym*V inplaceofT, Consider now themixed tensor Sym?V@V*.Itsweightsarethesumsof theweights ofSym? thatis,thepairwise sums oftheLywith theweights ‘ofV*;explicitly, these areLy+Ly—Lyand2L;~L;(each occurring, once) andtheL;themselves (each occurring three times, asL,+L,—L,). Dia grammatically, therepresentation looks like Now, weknow right offthebatthat this isnotirreducible: wehave anatural map aSym'V@ V8 siven again bycontraction, that is,bythemap ow@utaCo,ut)+diyut)0, whichisamapofslsC-modules.* WhatdoestheKernelofthismaplooklike?Ofcourse, itsweight diagram is *AnotherwaytoseethatSym?©¥*notIeeducbeitoobservethtifrepresentation Wis generatedbyahighestweighvectoroofweight2Ly~Ly.a8Sym!V@V*mostbeiti inreocibie, theeigenvalue can betaken with mullpiciy almost 2thecorresponding ‘genepace being generated byFi,2°F,s0and Ey Ey $131, Examples wt ‘and weknow one other thing: certainly any vector intheweight space of 2L,~Ly—that istosay,ofcourse, anymultiple ofthevector e?@e$—is killed BY1,14: Siy-tyy ANUGy, 80thatthekernel ofrwillcontain an irreducible representationF=I,with2L,—Lyasitshighestweight.Since T-must then assume every weight ofKer(i), there areexactly twopossibilities: ‘either Ker()) =P,which assumes theweights L,with multiplicity 2;orallthe weights ofFoccur with multiplicity one and Ker(i) =F@V. How dowesettle this issue? There areatleast three ways. Tobegin with, wecantrytoanalyze directly thestructure ofthekernel of1.Analternative approachwouldbetodetermineaprioriwithwhatmultiplicities theweights offyaretaken, Certainly itisclear that aformula giving usthelatter information willbetremendously valuable—it would foronething clear upthepresentconfusioninstantly—andindeedthereexistseveralsuch,oneofwhich, theWeyl character formula, wewill prove later inthebook. (We will alsoprove theKostant multiplicity formula, which canbeapplied todeduce directly theindependence statement wearrive atbelow.) Asathird possibility, wean identifytherepresentations F,,asWeylmodulesandappealtoLecture 6.Ratherthaninvokesuchgeneralformulas atpresent,however, wewilltakethefirst approach here. This isstraightforward: interms ofthenotation wehave been using, the highest weight vector for the representation TcSym?V@V*isthevectore?@ef,andsotheeigenspaceI,¢Twith cigenvalue L,willbespanned bytheimages ofthisvector under thetwocompositions E,, Es,yandEs,y©Ey...Theseae,respectively, Ba,©Es,2(0}@e$)=E2,1(Es,216}) @et+}@Es,2(¢$)) =Ex(-F @et) =—2e, 2)Bef+e}Bef and Es,20 Ba,s(e{ @e$)=Es,2(Es.s(€1) Beh+€7@Ey,sle8)) =By2ey-€2)@ 8) =eyes) @eF —2eye,)@et ie 15,Representations ofslyC,PartI:MainlyLotsofExamples Sincetheseareindependent, weconcludethattheweightLydoesoccurinwithmultiplicity 2,andhencethatthekernelofrisirreducible, ie. sym@Vt21,Ov §13.2. Description oftheIrreducible Representations Atthispoint, rather than goonwith more examples weshould state some of thegeneral principles that have emerged sofar.Thefirstandmost important {though pretty obvious) isthebasic Observation 13.2. Iftherepresentations VandWhave highest weight vectors and wwith weights «and f,respectively, then thevector v@we V@ Wis ‘ahighest weight vector ofweight «+P. Ofcourse, there arenumerous generalizations ofthis: thevector v*€Sym"V isahighest weight vector ofweight na,etc? Just thebasic statement above, however, enables ustogive the PROOF OFTHEOREM 13.1, Fits, theexistence statement follows immediatelyfromtheobservation: therepresentation Sym?V@Sym*¥*willcontainanirreducible subrepresentation I,with highest weight aL ~by ‘The uniqueness part isonly slightly harder (ifless explicit): Givenirreducible representations VandWwithhighestweighta,letveVandweWbehighest weight vectors with weighta.Then(0,w)isagainahighestweight vector intherepresentation V@Wwith highest weight a;let UcV@W betheirreducible subrepresentation generated by(r,»).The projection maps n,:UV and my:U—> W,being nonzero maps between irreduciblerepresentations ofsly,mustbeisomorphisms, andwededucethatV&W.a Exercise 13.3*, LetS,betheSchur functor introduced inLecture 6.What canyou sayabout thehighest weight vectors intherepresentation 5,(V) ‘obtained byapplying ittoagiven representation V7 Tocontinue ourdiscussion oftensor products like Sym*V@Sym*V*in ‘gencral,asweindicatedwewouldliketomakemoreexplicittheconstruction oftherepresentation TF, which weknow tobelying inSym*V@Sym*V*, Tobegin with, wehave ingeneral acontraction map ty:Sym*V@Sym?V*-+Sym?!¥@Sym?!V*analogous tothemap1introduced above;wecandescribethismapeither{infancy language) asthedual ofthemap from Sym*~! ¥@Sym?"!V*toSym*V@Sym*V* givenbymultiplication bytheidentityclementin 2OnetighlyesobviousstatementthitheweighsofVare, 4.2. with ay)>ey)> then/*¥posesehighestweightvectorweight,+--+,Rotethatsnestheondetng ‘ftv g;mayinfetJependonthechoiceof(evenwihtherestictona>b>onthecreficents ofavabovethiemayinsomecasesimplyteexistenceofseveralsubrepresentations of7, $132. Description ofthe treducible Representations 183 V@V* =Hom(¥, V);or,concretely, bysending (04-201) BOT 88) DipCTAee ReaD ‘Clearly thismap issurjective, and, since thetarget does nothave eigenvalue aL~bLy, thesubrepresentation T,,, ¢Sym*V@Sym*V*mustlieinthe kernel. Infact,wehave, justasinthecase ofSym?V@V*above, Claim13.4.Thekernelofthemap1,istheirreducible representation Ty. We will defer theproof ofthis for amoment and consider some ofits ‘consequences, Tobegin with, wecandeduce from thisassertion thecomplete decomposition ofSym*V@Sym*¥*: wemusthave(if,say.b<a) Sym*V@Sym'V*=@Tyiacr (13.5) ‘Sinceweknow,apriori,allthemultiplicities oftheeigenvalues ofthetensor product Sym"V@Sym*V®,thiswill,inturn,determine(inductively atleast) allthemultiplicities oftherepresentations T,,y. Infact, theanswer turns outtobevery nice. Toexpress it,observe first that ifa2h,theweight dia- gram ofeither T,,,,orSym*V @Sym*V’* looks likeasequence ofbshrinking ‘concentric (not ingeneral regular) hexagons #H,with vertices atthepoints (a—i)L, —(b—AL, fori =0, 1,...,6—|,followed (after theshorter three sides ofthehexagon have shrunk topoints) byasequenceof[(a~6/3]+1 triangles T;with vertices atthepoints (a—b—3/)L, forj=0, 1,...[a—by/3](itwillbeconvenient notationally torefertoT,asHyoccasionally),Diagram (13,6) shows thepicture oftheweights ofSym®V@Sym?V*: aN .~eNSNatyttySiyoty os (36) a 7 aeo. ee 184 1,Representations ofsC,Parttk:MainlyLotsofExamples (Note thatbythedecoinposition (13.5), theweights ofthehighest weight vectors inSym*V@Sym*V*willbeaL,—bLs,(a—I)Ly—(b—WLy,.--5 (a—b)Ly,a8showninthediagram.)Anexamination ofthe representationSym*V@Sym'V*showsthatithas multiplicity (7+ 1)(1+ 2/2 onthehexagon H,,and then aconstant muli- plicity (b+1)(b+2/2onallthetriangles 7};anditfollows from thedecom- position (13.5), ingeneral, that therepresentation Tyhasmultiplicity (i+ 1) ‘onHand bon‘In Englishthemultiplicities offincreasebyoneoneach oftheconcentric hexagons oftheeigenvalue diagram andareconstant onthe triangles. Note inparticular that thedescription ofT,, inthepreceding section isaspecial case ofthis. Proor oFCLAIM 13.4. Weremark first that theclaim willbeimplied bythe ‘Weyl character formula orbythedescription viaWeyl’s construction in Lecture 15;sothereader who wishes tocanskip thefollowing without dire consequences tothe logical structure ofthe book. Otherwise, observe first that theclaim isequivalent toasserting thedecomposition (13.5} this, intun, isequivalent tothestatement that the representation W=Sym*V@Sym'V*hasexactlyb+1irreducible components (still assuming a>b).The irreducible factors inarepresentation correspond tothehighest weight vectors intherepresentation uptoscalars; soin sum theclaim isequivalent tothe assertion that the eigenspace W,of Sym*V @Sym'V* contains aunique highest weight vector (uptoscalars) if«isoftheform(a—i)Ly—(b—i)Lyfori<b,andnoneotherwise; thisiswhat wweshall prove. ‘Tobegin with, the“none otherwise” part ofthestatement follows (given the other) just from looking atthediagram: if,forexample, any ofthe cigenspaces W,corresponding toapoint aonahexagon H;(other than the vertex (a—i)L, —(b —i)L ofH,)possessed ahighest weight vector, the multiplicity ofainWwouldbestrictlygreaterthanof(a—f)L,~(b—JL, which weknow isnotthecase; similarly, thefactthat themultiplicities ofW inthetriangular partoftheeigenvalue diagramareconstant implicsthatthere canbenohighest weight vectors witheigenvalue onaTJ;forj>|.Thus, we just have tocheck that theweight spaces W,fora= (a~i)Ly —(b =i)Ly contain only theonehighest weight vector weknow isthere; andwedothis byexplicit calculation ‘Tostart, foranymonomial indexI=(i,,i,,is)ofdegreeYi,=i,wedenote bye!«Sym'¥thecorresponding monomial ||(e})anddefine(e*)!€Sym'¥*similarly. Wecanthen write anyelement oftheweight space Wig-it,e-0y ‘ofSym*V @Sym'V* as w=Le lee @(eh! (ey). Imthese terms, itiseasy towrite down theaction ofthetwooperators E,. $13.3, ALittle More Plethysm iss and E,, First, E,, kills both e,¢Vand ef¢¥*,sothat wehave Eyee Olle He) =hfesheh@(esh (ery!) ile eVOUP) where I”=(i,+1,i;— 1,is)and I”=(iy—If,+1,is)(and weadopt the convention thate’=0if,<0foranyp)Itfollows thatthevector vabove isinthekernelofEy,ifandonlyifthecoefficients c,satisfyi,cy=(Iy+Yep:‘andbytheanalogous calculation thatvisinthekernel ofE,3ifandonlyif igcy=(iy+Neywhenever theindices Iand Jarerelated byjy=iy,Jy= i,+1, andJy=fy—1.These conditions areequivalent tosaying that the numbers i!f,tigley areindependent off.Wesee, inother words, that oisahighestweightvectorifandonlyifallthecoefficientsc,areequaltoc/i,lis!is! forsome constant ¢, oO §13.3. ALittle More Plethysm ‘Wewould liketoconsider here, aswedidinthecase ofsl,€ inLecture 11,howthetensorproducts andpowersoftherepresentations wehavedescribed decompose. Westart with one general remark: given our knowledge ofthe eigenvalue diagrams oftheirreducible representations ofsl,C (with multi- plicities) there canbenopossible ambiguity about thedecomposition ofanyrepresentation Ugivenasthetensorproductofrepresentations whosecigen-valuediagrams areknown.Indeed,wehaveanalgorithin fordetermining the components ofthat decomposition, asfollows: 1.Write down theeigenvalue decomposition ofU. 2.Find theeigenvalue «=aL, —bL, appearing inthisdiagram forwhich thevalue ofI)ismaximal 3.WenowknowthatUwillcontain acopyoftheirreducible representationT,=T,4,ie, U=T,@U'forsome U’.Since wealsoknow thecigenvalue diagram off,,wecanthuswritedownthecigenvalue diagramofU’aswell. 4,Repeat thisprocess forU’. ‘Tosechowthisgoesinpractice,considersomeexamplesoftensorproductsofthebasicirreducible representations describedsofar,Wehavealreadyseenhowthetensor products ofthesymmetric powers ofthestandard represen- {ation Vofs1,C andsymmetric powers ofitsdual decompose; jetuslook now atan example ofamoregeneraltensorproductofirreducible representations: sayVitselfandtherepresentation T,,.Westartbywritingdowntheweights ofthetensor product: since T,,hasweights 2L;—Ly,Ly+L; —Ly,andL, thanBethaynedagasnnthe(MERfourHimeshandaiy ATAVAVANY terefourseySepdeeonectroftheMase,andahelmlghswivetnowrightothebatthtthecnsorproduct@Fi,contanea VAVAVAVAVANLedeen.OeORD OO s0thecomplement ofFy,inthetensor product V@I,willlook like $13.3. ALittle More Plethysm 187 SERIOCODEAHOOOWAVAYA”AVATAYAYAY)WAVAYAA"AVATANOR” representation I,,aswell;sincethishasweightdiagram IINININININeeWAVAVACAN ALYYZ\REOK) ‘theremaining partofthetensor product willhave weight diagram IVAVAN vAARRERKERL\LX¥ OX)oeRRERYALY temorpuesVTV@T,,2andV@T5,;.Canyoufindageneral symmeticandesteroFTerentia terthanesandrthesymetticgunSyme?SymeSym),WeknowtheSaga Sym?W; itis LEERON seraitee cigenvave diagramboksfethis-wemus taseuauon wom §134. ALittle More Geometric Plethysm 189 ‘Sym?(Sym?) =Sym*V@Sym?V*. Indeed, thepresence oftheSym*V factor isclear: there isanobvious map «9:Symm*(Sym* V))+Sym* V obtained simply bymultiplying out.The identification ofthekernel ofthis ‘mapwiththerepresentation Sym?V*iscertainlylessobvious,bucanstillbe madeexplicit,WecanidentifyV*with\?Vaswesaw,andthendefineamap 2Sym3(\"¥) —Sym*(Sym?)) bysending thegenerator (uA 8)-(w A2)@Sym*(A*V) totheelement(w-w)-(0-2) —(wz):(0W)©Sym?(Sym?V),whichisclearlyinthekernelof¢. Exercise 13.9. Verily that thismapiswell defined andthat itextends linearlyto.anisomorphism ofSym?(A*V) withKer(p). Exercise 13.10. Apply thetechniques above toshow that therepresentation, A*(Sym?V)isisomorphic toT, Exercise13.11.Applythesametechniques todeterminetheirreduciblefactors oftherepresentation A?(Sym?V). Note: wewillreturn tothisexample in Exercise 13.22. Exercise 13.12. Find thedecomposition into irreducibles oftherepresenta- tions Sym?(Sym? V)andSym*(Sym?V)(observeinparticularthatHermite reciprocity hasbitten thedust). Describe theprojection maps tothevariousfactors.Note:wewilldescribetheseexamplesfurtherinthefollowingsection. §13.4. ALittle More Geometric Plethysm Just asinthecase ofsI3C, some ofthese identifications canalso beseen in geometric terms. Todothis, recall from §11.3 thedefinition oftheVeroneseembedding: ifP?=PV*istheprojectivespaceofone-dimensional subspacesofV*,there isthen anatural embedding ofP?intheprojective space P*= P(Sym?V*), obtained simply bysending thepoint (o"] €P®correspondingtothevectoro*€V*tothe point [o®#] €P(Sym?V*) associatedtothevector of?=otoeSym?V*. Theimage 5cP*iscalled theVeronese surface. As inthe case ofthe rational normal curves discussed inLecture 11,itis not hard {0seethatthegroup ofautomorphisms ofP*carrying Sintoitselfisexactly thegroupPGLy€ofautomorphisms of5=P#Now,aquadraticpolynomial inthehomogeneous coordinatesofthespace P(Sym?V*) xP°willrestrict toquartic polynomialontheVeronesesurface 5=PV®,whichcorresponds tothenaturalevaluation mapgofthepreceding section; thekernel ofthis map isthus thevector space ofquadratic poly- 190 15,Representations of1,€,PartI:MainlyLotsofExamples rnomials inP?vanishing ontheVeronese surface S,onwhich thegroup of automorphisms ofPcarryingStoitselfobviouslyacts.Now,foranypairof pointsP=(u*],Q=[oP]€5,itisnothardtoseethattheconeovertheVeronesesurfacewithvertexthelinePQ<P(thatis,theunionofthe2-planes POR asRvaries over thesurface S)willbeaquadric hypersurfaceinP*containing theVeronesesurface;sendingthegeneratoru*-v*€Sym?V* tothisquadric hypersurface willthen define anisomorphism ofthespace of such quadrics withtheprojective space associated toSym? V*, Exercise 13.13. Verify thestatements made inthelastparagraph: thatthetunionofthePQRisaquadrichypersurface andthatthisextendstoalinearisomorphism P(Sym?V*)xP(Ker(9)).Verifyalsothatthisisomorphism coincides with theone given inExercise 13.9. ‘There isanother way ofrepresenting theVeronese surface that willshed some light onthiskernel. If,interms ofsome coordinates e,onV*,wethinkofSym*V*asthevectorspaceofsymmetric 3x3matrices,thentheVeronese surfaceisjustthelocus,intheassociatedprojectivespace,ofrank|matrices ‘uptoscalars, ie,interms ofhomogeneous coordinates Z,,=e,"¢, 00P*, Zia Zy2 Zs)s-faontZtze)eal. Zs Zrs Zss ‘Thevectorspaceofquadratic polynomials vanishingonSisthengeneratedbythe2x2minorsofthematrix(Z,inparticular, foranypairoflinearcombinations oftherows andpair oflinear combinations ofthecolumns we geta2x2mattix whose determinant vanishesonS. Exercise 13.14. Show that this isexactly theisomorphism Sym?(\?V)= Ker(o) described above. ‘Wenote inpassing that ifindeed thespace ofquadrics containing the Veronese surface, with theaction ofthegroup PGL3€ ofmotions ofP*preserving 5,istheprojectivization oftherepresentation Sym?V*,thenitmustcontain itsown Veronese surface, ie,there must beasurface T=P(V*)< P(Ker(o)) invariant under this action, This turns out tobejust thesetof quadricsofrank3containingtheVeronese,thatis,thequadricswhosesingular locus isaplane. Infact,the2-planewillbethetangentplanetoSatapoint, giving the identification T=S. Letusconsideronemoreexampleofthistype,namely,thesymmetriccube ‘Sym*(Sym?V)).(Wepromisewewillstopafterthisone.)Asbefore,itiseasy towrite down theeigenvalues ofthisrepresentation; they arejust thetriplesumsoftheeigenvalues {2L,, L,+Ly}ofSym?¥V.Thediagram(wewilldraw here only one-sixth oftheplane and indicate multiplicities with numbers rather than tings) thus looks like §134. ALittle More Geometric Pethysm 1 from which weseewhat thedecomposition must be:asrepresentations we have Sym?(Sym?) =Sym°V@12,2. (13115) ‘Asbefore, themap tothefirst factorisusttheobviousone;itistheidentifica- tionofthekernelthatisintriguing, andespeciallytheidentification ofthelast factor. Toseewhat isgoing onhere, weshould look again atthegeometry ofthe Veronese surface $<P*=P(Sym*V*), Thespace Sym°(Sym?V))isjustthe space ofhomogeneous cubic polynomials ontheambient space PS,andasbeforethemaptothefirstfactoroftheright-hand sideof(13.15) isjust the restriction, sothat thelast two factors of(13.15) represent thevector space (S)3 ofcubic polynomials vanishing onS.Note that wecould infactprove (13.15) without recourse toeigenvalue diagrams from this: since theideal of theVeronese surface isgenerated bythevector space I(S), ofquadratic polynomials vanishing onit,wehave asurjective map US. W=Sym2V*@Sym?V—+1(5)s. Butwealready know how thelefthand sidedecomposes: wehave Sym?¥*@Sym?V=13,21), OC, (1316) $0that1(S)mustbeapartialdirectsumofthesethreeirreducible represen-tations;bydimension considerations itcanonlybeI,©€.This,inturn,tellsushowtomaketheisomorphism (13.15)explicit(assum-ingwewant to}:wecan define amap Sym?(\? V)@Sym?V+Sym?(Sym?V) bysending, (A904 2) (8-H (GUW)(02) —w-2)-(0- 1)(9) 12 13,Representations ofsy€,PartMH:MainlyLotsofExamples and then just check that this gives anisomorphism of13,2 ¢ Sym?V*@Sym?Vwiththekernelofprojection onthefirstfactoroftheright-hand sideof(13.15). ‘What isreally most interestinginthiswholesituation,though,isthetrivial summandintheexpression (13.15),Tosaythatthereissuch@summandisto saythat there exists acubic hypersurface XinP®preserved under allanto- ‘morphismsofP®carryingStoitself.Ofcourse,wehavealreadyrunintothis one: itisthedeterminant ofthe3x3matrix (Z,,) introduced above. To ‘express thismore intrinsically, ifwethink oftheVeronese asthesetofrank {tensorsinSym?V4,itisjustthesetoftensorsofrank 2orless. This, inturn, yields another description ofX:since arank 2tensor isjustonethat canbe ‘expressed asalinearcombination oftworankItensors,weseethatXisthe Famous chordal variety oftheVeronese surface: itistheunion ofthechordstoS,andatthesametimetheunionofallthetangentplanestoS. Exercise 13.17. Show thattheonly symmetric powers ofSym?V thatpossesstrivialsummands arethepowersSym*(Sym?)) divisibleby3,andthatthe‘unique trivial summand inthisisjustthekthpower ofthetrivial summand ofSym*(Sym*V)). Exercise 13.18. Given theisomorphism oftheprojectivization ofthevector space I(S),—that is,theprojective space ofquadric hypersurfaces containing theVeronese surface—with P(Sym*V*), findtheunique cubic hypersurface in1(S), invariant under theaction ofPGL,C. Exercise 13.19. Analyze therepresentation Sym?(Sym*V)) ofsl,C. Interpret thedirect sum factors intermsofthegeometryoftheVeroneseembeddingof PV* =P?inP(Sym'V*)=PP. Exercise 13.20, Show that the representations Sym‘(Sym?¥)) and Sym®(Sym°¥)) contain trivial summands, and that the representation Sym'(Sym'V)) contains two. Interpret these. Exercise 13.21. Apply thetechniques above toshow that therepresentation (Sym? V)isisomorphic toT, Exercise 1322*. Apply thetechniques above toanalyze therepresentation /%(Sym*V), andinparticular tointerpret itsdecomposition intoirreducible representations. Exercise 13.23. IfP*=P(Sym?V*) istheambient space oftheVeronese surface, theGrassmannian G(2, 5)of2-planes inPSnaturally embeds inthe projective space P(A*(Sym*V)). Describe, interms ofthedecomposition imthepreceding exercise, thespan ofthelocus oftangent 2-planes tothe $134. ALite Mote Geometric Plethysm 193 Veronese,andthespanofthelocusof2-planesinP®spannedbytheimagesin$oflines inPY, Exercise 13.24*. Show thattheunique closed orbit oftheaction ofSLsC ontherepresentation I,iseitherisomorphic toP?(embedded astheVeronesesurface) ifeitheraorbiszero,ortotheincidencecorrespondence Em((p.):pel) cP? xPY itneither aorbis zero. ° PART III THE CLASSICAL LIE ALGEBRAS AND THEIR REPRESENTATIONS ‘Asweindicated attheoutset, theanalysis wehave just carried out ofthe structure ofstC and slyC and their representations carries over toothersemisimple complexLiealgebras,InLecture14wecodifythisstructure,usingthepattern oftheexamples wehave worked outsofartogive amodel fortheanalysisofarbitrarysemisimple Liealgebrasandstatingsomeofthemost, important facts thataretrueingeneral. Asusual, wepostpone proofsofmany ofthese facts until Part 1VandtheAppendices, themain point here being to introduce @unifying approach andlanguage. Thefacts themselves willallbeseenexplicitlyonacase-by-case basisfortheclassicalLiealgebras#l,C,sp,C,ands0,C,whicharestudiedinsomedetailinLectures15-20.Most ofthedevelopment follows theoutline wedeveloped inLectures 11-13, themain goal being todescribe the irreducible representations as explicitly aswecan, and t0seethedecomposition ofnaturally occurring representations, both algebraically and geometrically. While most oftherepresentations arefoundinsidetensorpowersofthestandardrepresentations,fortheorthogonal Liealgebras this only gives half ofthem, and one needs new methods toconstruct theother “spin” representations. This carried out using Clifford algebras inLecture 20. Wealso make thetiewith Weyls construction ofrepresentations of GLC. from Lecture 6,which arose from therepresentation theory ofthesymmetric groups. Weshow inLecture 15that these aretheirreducible representationsofsl;inLecture17weshowhowtousethemtoconstructtheirreduciblerepresentations ofthesymplectic Liealgebras, and inLecture 19togive the nnonspin representation oftheorthogonal Liealgebras. These give useful descriptions oftheirreducible representations, and powerful methods for decomposing other representations, butthey arenotnecessary forthelogical progression ofthe book, and many ofthese decompositions can also be deduced from theWeyl character formula which wewilldiscuss inPart IV. LECTURE 14 The General Setup: Analyzing the Structure and Representations ofan Arbitrary Semisimple LieAlgebra ‘Thisisthefastofthefourcentrallectures; inthebodyofit,§14.1,weextractfromthe‘examples of§1!—13thebasicalgorithm foranalyzing ageneralsemisimple Liealgebranditsrepresentations. isthisalgorithmthatwewlspendtheremainderofPartTHcarrying outfortheclassical algebras, andthereader who finds thegeneral setupconfusingmaywishtoreadthlseetureinparallelwithforexample,Lectures15and16.tnparticular, §142 islessclearly motivated bywhat wehave worked out80fr; therealer may wish(o skim iformow and defer amore thorovgh reading til atersingthroughsomemoreoftheexamplesofLectures15-20 §14.1: Analyzing simple Liealgebras ingeneral. §14.2: About theKilling form §14.1. Analyzing Simple LieAlgebras inGeneral ‘WessaidattheoutsetofLecture12thatoncetheanalysisoftherepresentations. ofl,€wasunderstood, theanalysisoftherepresentations ofanysemisimpleLiealgebra would beclear, atleast inbroad outline, Here wewould liketo indicate how that analysis willgoingeneral, byproviding anessentially algorithmic procedure fordescribing therepresentations ofanarbitrary com-plexsemisimple Liealgebra9,Theprocesswegivehereisdirectlyanalogous,step forstep, tothat carried outinLecture 12forsl,€; theonly difference is‘onechangeintheorderofsteps:havingseeninthecaseofsl€theimportance ofthe“distinguished” subalgebras s,=sl;€ <gandthecorresponding dis- tinguished elements H,€,<b,wewillintroduce them earlier here. Step 0.Verify that your Liealgebra issemisimple; itnot, none ofthe following willwork (but seeRemark 14.3). Ifyour Liealgebra isnotsemi- simple, pass asindicated inLecture 9toitssemisimple part; aknowledge of therepresentations ofthisquotientalgebramaynottellYoueverything about 198 14,TheGeneralSet-up:AnalyzingtheStructure therepresentations oftheoriginal, butitwill atleast tellyou about the irreducible representations. ‘Step I.Find anabelian subalgebra ycgacting diagonally. This isofcourse theanalogue oflooking atthespecific element Hinsl,C andthesubalgebra}ofdiagonal matrices inthecase ofsly€; ingeneral, toserve ananalogousfunctionitshouldbeanabeliansubalgebra thatactsdiagonally ononefaithful(and hence, byTheorem 9.20, onany) representation of9,Moreover,inorder that the restriction ofarepresentation Vofgtohcarrythegreatestpossible information about ¥V,)should clearly bemaximal among abelian, diagonali- zable subalgebras; such asubalgebra iscalled aCartan subalgebra. ‘Note thatwhile thisstepwould seem tobesomewhat lessthan algorithmic(inparticular, whileitiscertainlypossibletotellwhenasubalgebra ofagivenLicalgebra isabelian, andwhen itisdiagonalizable, itisnotclear how totell whether itismaximal with respect tothese properties). This defect will, however, belargely cleared upinthenext step (see Remark 14.3). Step 2.Let hactongbytheadjoint representation, and decompose 9 ‘accordingly. Bythechoice off, itsaction onanyrepresentation ofgwillbe<iagonalizable; applyingthistotheadjointrepresentation wearriveatadirectsuum decomposition, called aCartan decomposition, =bO@a) (14.1) where theaction ofl preserves each 9,and acts onitbyscalar multiplication bythelinear functional a-€5;that is,foranyHehand Xeg,wewillhave ad(H)(X)=a(t)X. ‘Theseconddirectsumintheexpression(14.1)isoverafinitesetofeigenvalues aeh*;these cigenvalues—in thelanguage ofLecture 12,theweights ofthe adjoint representation—are called therootsoftheLiealgebraandthecorre- spondingsubspaces g,arecalledtherootspaces.Ofcourse,itselfisjustthecigenspace fortheactionofhcorresponding totheeigenvalue 0(sceRemark14.3below); sothat insome contexts—such asthefollowing paragraph, for example——it willbeconvenient toadopt theconvention that g~b;butwe donotusually count 0.h*asaroot. The setofallroots isusually denoted Robe. Asintheprevious cases, wecanpicture thestructure oftheLiealgebra intermsofthediagramofitsroots: bythefundamental calculation of §11.1and Lecture 12(which wewillnotreproduce here forthefourth time) weseethat theadjoint action ofacarries theeigenspace gyintoanother eigenspace gasp.‘Thereareacoupleofthingswecananticipateabouthowtheconfiguration ofroots (and thecorresponding root spaces) will look. Wewillsimply state them here as Facts 142 (i)each root space gywilbeonedimensional.Gi)RwilgeneratealatticeAy&h*ofrankequaltothedimensionofb. S141, Analyzing Simple LieAlgebras inGeneral 199 (ii)Rissymmetric ahout theorigin, Le,fa€Risaroot, then —we Ris root aswell. ‘These facts willallbeproved ingeneral induecourse; forthetime being,theyarejustthingswewillobserveaswedotheanalysisofeachsimpleLiealgebra inturn, Wemention them here simply because some ofwhat follows willmake sense only given these facts. Note inparticular that by(i),theroots ailfiein(and span)arealsubspaceofh*;allourpicturesclearlywillbeofthis realsubspace. Remark 14.3 Ifindeed 0does appear asaneigenvalue oftheaction of hon a/b,then wemay conclude from thisthat }wasnotmaximal tobegin with:bytheabove,anythingintheO-cigenspace oftheactionofhcommutes with‘hand (given thefactthat theg,areone dimensional) acts diagonally on9,30 that ifitnotalready inb,then could beenlarged while still retaining the properties ofbeing abelian and diagonalizable, Similarly, theassertion in(i) that theroots span *follows from thefact that anelement of inthe annihilator ofallofthemwouldbeinthecenterof9, From what wehave done sofar,wegetourfirstpicture ofthestructure ofanarbitraryirreducible finite-dimensional representation Vofg.Specifically,Vwill admit adirect sum decomposition v=O% (144) wherethedirectsumrunsoverafinitesetof«€h*andbyactsdiagonally oneachV,bymultiplication bytheeigenvalueaie,foranyH€handveVzwe will have HG) =a(tty-v. ‘Theeigenvalues ¢b*thatappearinthisdirectsumdecomposition arecalledtheweights ofV;theV,themselves arecalled weight spaces, andthedimension ‘ofaweightspaceV,willbecalledthemultiplicity oftheweightainV.Wewilloften represent Vbydrawing apicture ofthesetofitsweights andthinking ofeach dotasrepresenting asubspace; this picture (often with some annota-tiontodenotethemultiplicity ofeachweight)iscalledtheweightdiagramofV.‘The action oftherest oftheLiealgebra onVcan bedescribed inthese terms: forany root f,wehave 82Vat Voss s0wecanthinkofthe action ofg,onVasatranslation inthe weight diagram, shifting each ofthedots over byfand mapping the weight spaces correspondingly. Observe next that alltheweights ofanirreducible representation are congrtient tooneanother modulo theroot lattice Ay:otherwise, forany weight «ofVthesubspace 20 14,TheGeneralSet-up:AnalyzingtheStructure v=@Yor wouldbeapropersubrepresentation ofV:Inparticular,in viewofFact14.2),thismeans thattheweights ale inatranslate ofthe realsubspace spanned bytheroots,s0thatitisnotsounreasonable todrawapictureofthem. Step3.Findthedistinguished subalgebras 5,%sl,<g.Aswesawinthe‘exampleofsy€,acrucialingredient intheanalysisofanarbitraryirreduciblefinite-dimensional representation istherestriction oftherepresentation to certain special copies ofthealgebra sl,contained ing,andtheapplication ofwhat weknow from Lecture 11about such representations. TogeneralizethistoourarbitraryLiealgebra9,letgg€9bearootspace,onedimensionalby(i)ofFact14.2,Thenby(ii)ofFact14.2,thereisanotherrootspace9»€ andtheir commutator [a4,9-.] must beasubspace ofgo=b,ofdimension‘atmostone.Theadjointactionofthecommutator [4,,6]thuscarrieseach‘ofgqand g.-into itself; sothat thedirect sum $2=Ge®9-2[Ge9-2) (14.5) isasubalgebra of9,The structure ofs,isnothard todescribe, given two further facts that wewillstate here, verily incases, and prove ingeneral in Appendix D. Facts 14.6. ([ae8-2] #0;and Gi)(LG, 9-1) 96]#0. Given these, itfollows that thesubalgebra s,isisomorphic tosly. In particular, wecanpick abasis X,©dy»Ys€Geand Hy€[Oy8-r] satislying thestandard commutation relations (9.1) forsl,€; X,and ¥,arenotdeter- mined bythis, butH,is,being theunique element of[jq,@-«) having eigen-values2and~2ong,andg..,respectively (ie,H,isuniquelycharacterizedbytherequirements that Hy€ [88-»] and a(H,) =2] Step 4.Usetheintegrality oftheeigenvalues oftheHy.Thedistinguished elements H,€bfound above areimportant firs ofall because, bytheanalysisoftherepresentations ofs|,€ carried outinLecture 9,inany representation ofs,—and hence inanyrepresentation ofg—all elgenvalues oftheaction of H,must beintegers. Thus, every eigenvalue f€h*ofevery representation of {9must assume integer values onalltheH,.Wecorrespondingly letAybe thesetoflinear functionals fieb*thatareinteger valued onalltheHy;Awwillbealattice,calledtheweightlatticeofg,withthepropertythatallweightsofallrepresentations of9willieinAy. Note, inparticular, that R&C Ayand hence Ag&Aw; infact, theroot lattice willingeneral beasublattice offinite index intheweight lattice.Step5.Usethesymmetryoftheeigenvalues oftheH,.Theintegrality ofthe $14.1. Analyzing Simple LieAlgebras inGeneral 201 eigenvalues oftheH,under anyrepresentation isonly halfthestory; itisalso true that they aresymmetric about theorigin inZ.Toexpress this, forany weintroduce theinvolution W,onthevector space h*with +I-cigenspace thehyperplane 0,=(BEY: (HyBD=0} (14.7) andminus|eigenspace thelinespanned by«itself.’InEnglish, W,isthe Teflection intheplane Q,with axis thelinespanned bya: 2p wep)=p—Aled=p—pitteyx (148) a(H,) ‘LetWBbethegroupgenerated bytheseinvolutions; WiscalledtheWeylgroupoftheLiealgebra g,‘NowsupposethatVisanyrepresentation ofg,witheigenspace decomposi-tion¥=G)%.Theweightsfappearinginthisdecomposition canthenbebroken upintoequivalence classes mod a,andthedirect sum Yn@Yow (149) oftheeigenspaces inagiven equivalence class willbeasubrepresentation of Vfors,.Itfollows then that thesetofweights ofVcongruent toanygiven {fmod &willbeinvariant under theinvolution W;inparticular, Thesetofweights ofanyrepresentation ofgtsinvariant under theWeyl rou. ‘Tomakethismoreexplicit,thestringofweightsthatcorrespond tononzeto summands in(149) are,possibly afer replacing fbyatranslate byamultiple ote BB+9B422.0.5fmmwithm=—BUH,)(14.10) (Notethatbyouranalysisofst,€thismustbeanuninterrupted string)Indeed itwechoosefandm>Osothat(I4.10)sthestringcorresponding tononzero summands in(14.9), then thestringofintegersOHAB+9H)=Bt)+2,(+ma),)=AH)+2m ust besymmetric about zero, s0f(H,) =—m. Inparticular, WD+ka)=f+(—PUH,)—Kya=f+(m—Ka Note also that bythesame analysis themultiplicities oftheweighs are invariant under theWeyl group. Weshould mention oneother factabout theWeyl group, whose proof we: also postpone "Not thtbythe ondgeeacy ern ie Fat 146 thete dc ote inthe Iyer, Real nt) tepig beeen ed) =A) 202 14.TheGeneralSet-up:AnalyzingtheStructure Fact (4.11, Every element oftheWeyl group isinduced byanautomorphism oftheLiealgebra gcarrying \toitself. Wecanevensaywhatautomorphism ofqdoesthetrick:togettheinvolutionW,,take theadjoint action oftheexponential exp(niU,) €G,where GisanygroupwithLiealgebraqandU,is@suitableelementofthedirectsumofthe root spacesg,andg-,..ToprovethatAd(exp(iU,)) actuallydoesthisrequires moreknowledge ofqthanwecurrentlypossess;butitwouldbeanexcellentexercise toverify this assertion directly ineach ofthecases studied below. (For thegeneral case see(23.20) and(26.15}) Step 6.Draw thepicture (optional). While there isnological need todoso atthispoint, itwillbemuch easier tothink about what isgoing oninb*if weintroduce theappropriate inner product, called theKilling form, ong (hence byrestriction onb,and hence on). Since theintroduction ofthe Killing form is,logically, adigression, wewill defer until later inthislecture adiscussion ofitsvarious definitions and properties. Itwillsuffice fornow (0 ‘mention thecharacteristic property oftheinduced inner product onb*:upto scalars itistheunique inner product onh*preserved bytheWeyl group, ie,intermsofwhichtheWeylgroupactsasagroupoforthogonal transforma-tions. Equivalently, iistheunique inner product (uptoscalars) such that the Jinespanned byeach root a€bis actually perpendicular totheplaneM,(so that theinvolution W,isjust areflection inthat hyperplane). Indeed, in practice thisismost often how wewillcomputeit.Inermsofthe Killing form, then, wecansaythat theWeyl group isjust thegroup generated bythereflections inthehyperplanes perpendicular totherootsoftheLiealgebra.Step7.Chooseadirectionin*.Bythiswemeanarealfinearfunctional! conthelattice Ayirrational with respect tothislattice. This gives usa decomposition oftheset R=RUR, (14.12) whereR*=(a:(a)>0}{theaeR*arecalledthepositiveroots,thoseinR~negative}; this decomposition iscalled anordering oftheroots. For most purposes, theonly aspect of!thatmattersistheassociatedorderingofthe roots. Thepoint ofchoosing aditection—and thereby anordering oftheroots R=R*URIs, ofcourse, tomimicthenotionofhighestweightvectorthat ‘was50crucial inthecases ofsl, ands1,€. Specifically, wemake the Definition, LetVbeanyrepresentation ofg.Anonzero vector v€Vthatis both aneigenvector fortheaction ofhandinthekernelofgyforallaeR* iscalled ahighest weight vector ofV. Justasinthepreviouscases,wethenhave Proposition 14.13. Foranysemisimple complex Liealgebra 9, (i)every finite-dimensional representation Vofqpossesses ahighest welght vector; $14.1. Analyzing Simple LieAlgebras inGeneral 201 (id)thesubspace Wof¥generated bytheimages ofahighest weight vector » under suecessive applications ofrootspaces gyforB€Risanirreducible subrepresentation;(ii)amirreducible representation possessesauniquehighestweightvectorupo scalars. Proor. Part (i)isimmediate: wejusttake atobetheweight appearing inV forwhich thevalue (a)ismaximal and choose vany nonzero vector intheweightspace¥,-SinceV,.4=(0)forallfR*,suchavectorowillnecessarily beinthekernel ofall root spaces gycorresponding topositive roots f.Part(i)maybeprovedbythesameargumentasinthetwocaseswehavealready discussed: weletW,bethesubspace spanned byallw,-0where is8‘wordoflengthatmostminelementsofg,fornegativef.WethenclaimthatforanyXinany positive rootspace, X«W,<W,,Tosee this,writes generatorofW,intheform¥-w,w€W,..,,and usethecommutation relationX=Y-w=Y-X-w +[X,¥]-w; theclaim follows byinduction, since (X,YJisalways inb,The subspace Wc Vwhich is«union ofallthe1, isthus asub- representation; toseethatitisirreducible; note thatifwe write W=W"@ W", then either "or W*will have tocontain theone-dimensional weight space W,and sowillhave toequal W.‘Theuniqueness ofthehighestweightvectorofanirreduciblerepresentation followsimmediately: if»€V,andwe¥,weretwosuch,notscalarmultiples ‘ofeach other, wewould have I(2)>1(f) and vice versa. fa} Exercise 14.14, Show that in(i)oneneed only apply those gyforwhich‘y'#0.(Note:withW,definedusingonlythesegp,andXinanyrootspace,thesame inductive argument shows that XW, cW,,,. Ontheother hand,ifoneusesallgywithflnegativeandprimitive,asinObservation 14.16,thenX-W,<W,-.,Onecannotcombinethese,however:Vmaynotbegeneratedbysuccessively applyingthoseg,withfnegative,primitive,andgy-v#Oe8.,thestandardrepresentation ofsly.) ‘The weight «ofthehighest weight vector ofanirreducible representation willbecalled, notunreasonably, thehighest weight ofthatrepresentation; the {erm dominant weight isalso common. Wecanrefine part(ji)ofthisproposition slightly inanother direction; this, isnot crucial butwillbeuseful later oninestimating multiplicities ofvarious. tepresentations. This refinement isbased on Exercise 14.15*. (a)Let4,....a4beroots ofasemisimpleLiealgebraqand 4, <qthecorresponding root spaces. Show that thesubalgebra ofggene-ratedbytheCartansubalgebra btogetherwiththeq,isexactlythedirectsum ®(a),wherethedirectsumisovertheintersection ofthesetRofroots ofgwith thesemigroup N{ay,..-.4%) <bgenerated bythea.(b)Similarlyletay...benegativerootsofasemisimple Liealgebraa‘and9,,©9thecorresponding rootspaces.Showthatthesubalgebra ofggene- 204 14.TheGeneralSet-up:AnalyzingtheStructure ratedbytheg,,isexactlythedirectsum€P9,wherethedirectsumisovertheintersection ofthesetRofroots ofqwith thesemigroup N{a.....%4} <0 generated bythea(Notethatbythedescription oftheadjointactionofaLiealgebraonitself wehave anobvious inclusion; theproblem here istoshow—given thefacts above—that if+BeR,then fo,0)]#0) From thisexercise itisclear that generating asubrepresentation WYofa given representation Vbysuccessive applications ofrootspacesyfor'&R™ to.a highest weight veclor 0isinefficient; weneed only apply theroot spaces {yCorresponding t0asetofroots fgenerating R™asasemigroup. Weaccordingly introduce anotherpieceofterminology: wesaythatapositive(esp,negative)roota€Risprimitiveorsimpleifitcannotbeepxressedasasum oftwopositive (resp. negative) roots. (Note that, since there areonly finitely many roots, every positive rootcanbewritten asasumofprimitive positive roots) Wethen have Observation 14.16. Anyirreducible representation Visgenerated bytheimages Ofitshighest weight vector vunder successive applications ofroot spaces gy where firanges over theprimitive negative roots. Wehave already seen oneexample ofthis inthecase ofsly, where we observed (intheproofofClaim12.10andintheanalysisofSym?V@V*in Lecture13)thatanyirreduciblerepresentation wasgeneratedbyapplyingthe twoelements E2,1 ©Gc.-1, andEs,2 €Buy-t, t0ahighest weight vector.Toreturntoourdescriptionoftheweightsofanirreduciblerepresentation V,weobservenextthatinfacteveryvertexoftheconvexhulloftheweights ofVmustbeconjugatetoaundertheWeylgroup.Toseethis,notethatbythe above thesetofweights iscontained inthecone a+C;,where C,isthe positive realcone spanned bytheroots feR~such thatag(o) #0—that is, such thata(H,) #0.Conversely, theweights ofVwillcontain thestring of weights 0+ByBonn+(OCH,)IP (aan foranyfl€R~.Thus, anyvertex oftheconvex hullofthe setofweights of¥ adjacent toamust beoftheform a—a(H,)p =Wyle) forsome fl;applying thesame analysis toeach successive vertex gives the statement, Fromtheabove,wededucethatthesetofweightsofVwillfiintheconver.‘hulloftheimagesofaundertheWeylgroup.Since,moreover,weknowthat theintersection ofthissetwithanysetofweightsoftheform(f+ny}willbe‘connected string, itfollows that thesetofweights ofVwillbeexactly the weights that arecongruent to&modulo theroot lattice Axandthat lieithe convex hulloftheimages ofaunder theWeyl group. S14.1. Analyzing SimpleLieAlgebrasinGeneral 205 Onemorebitofterminology, andthenwearedone.Bywhatwehaveseen (cf.(14.17), thehighest weight ofanyrepresentation ofVwillbeaweight « satisfying a(H,) 20forevery y©R*.Thelocus ¥,,intherealspan ofthe roots, ofpoints satisfying these inequalities—in terms oftheKilling form, making anacute orright angle with each ofthepositive roots—is called the(closed)Weylchamberassociated totheorderingoftheroots.AWeylchamber ‘could also bedescribed astheclosure ofaconnected component ofthecomplement ofthe union ofthehyperplanes 0,.The Weyl group acts simply transitively onthesetofWeyl chambers andlikewise onthesetoforderings‘oftheroots.Asusual,thesestatements willbeeasytoseeinthecaseswestudy, while theabstract proofs arepostponed (toAppendix D). Step 8.Classify theirreducible, finite-dimensional representations of9 Where alltheabove isleading should bepretty clear, itisexpressed inthe fundamental existence anduniqueness theorem: ‘Theorem14.18,ForanyaintheintersectionoftheWeylchamberWassociated totheordering oftheroots with theweight lattice Aw, there exists aunique irreducible, finite-dimensional representation I,ofqwith highest weight a;this, gives abijection between WM Ayandthesetofirreducible representations of 8,TheweightsofIwillconsistofthoseelementsoftheweightlatticecongruent toamodulotherootlatticeAgandlyingintheconvexhullofthesetofpoints in9*conjugate toaunder theWeyl group. Hate-rroor. Wewill give here just theproof ofuniqueness, which iseasy.‘Theexistencepartwewilldemonstrate explicitlyineachexampleinturn;andlateronwewllsketchsomeoftheconstructions thatcanbemadeingeneral The uniqueness part isexactly thesame asforsly. IfVand Waretwo irreducible, fnite-dimensional representationsofgwithhighestweightvectors ‘and w,respectively, both having weight a,then thevector (s,w)€V@ W willagain beahighest weight vector ofweight ainthat representation. Let Uc V@W bethesubrepresentation generated by(0,w);since Uwillagain beirreducible theprojection mapsn,:U-»Vandx:U+W,beingnonzero, willhave tobeisomorphisms. a ‘Another factwhich wewillseeaswegoalong—and eventually prove in general—is that there arealways fundamental weights 02, ....@,with the Property that any dominant weight canbeexpressed uniquely asanon- ‘negative integral linear combination ofthem. They can becharacterized geometrically asthefirst weights met along theedges oftheWeyl cham- ber,oralgebraically asthose elements «in h*suchthat«n(H,,) =&,,,,where 4,-..,%arethesimple roots (insome order). When wehave found them, weoften write T,,,.._., fortheitreducible representation with highest weight 604 $0" +Gorey Tess =Taynybosay ‘Aswith most ofthematerial inthis section, general proofs will befound in Lecture 21and Appendix D. 206 14,TheGeneralSet-up:AnalyzingtheStructure ‘One basic point wewant torepeat here (and that wehope todemonstrate insucceeding lectures) isthis:thatactually carrying outthisprocessinpracticeiscompletely elementary andstraightforward. Any mathematician, stranded ‘onadesert island with only these ideas and thedefinition ofaparticularLie algebra gsuch as¢1,C, s0,C, orsp,,C, would inshort order have acompletedescription ofalltheobjectsdefinedaboveinthecaseofg.Weshouldsayaswell, however, thatattheconclusion ofthisprocedure weareleftwithout one vital piece ofinformation about therepresentations ofg,without which we willbeunable (oanalyze completely, forexample, tensor products ofknown representations; thisis,ofcourse,adescription ofthemultiplicitiesofthebasic representations I.Aswesaid, wewill, infact, describe and prove such a formula (the Weyl character formula); butitisofamuch lessstraight- forward character (our hypothetical shipwrecked mathematician would have tohave what could only bedescribed asapretty good daytocome upwith thedea) and willbeleftuntil later. Fornow, wewillconclude thislecture with thepromised introduetion totheKilling form §14.2. About theKilling Form {Aswesaid, theKilling form isaninner product (symmetric bilinear form)on theLiealgebra g;abusing ournotation, wewilldenote byBboth theKilling formandtheinduced innerproducts onbandh*.Bcanbedefined inseveral‘ways; themost common isbyassociating toapair ofelements X,Yegthe trace ofthecomposition oftheir adjoint actions ong,ic, B(X, ¥)=Tr(ad(X) ©ad(¥): g-+9). (14.19) ‘Aswewillsee,theKillingformmaybecomputed inpractice eitherfromthisdefinition, or(up toscalars) byusing itsinvariance under the group of automorphisms ofg.Weremarkthatthisdefinition isnotasopaque asitmayseem atfirst. Forone thing, thedescription oftheadjoint action oftheroot space g,asa“translation” oftheroot diagram—that is,carrying each root space gyintog,4p—tells usimmediately thatg,isperpendicular toggforall Plother than —aj inother words, thedecomposition o-50(@ 6.09.0) (14.20) isorthogonal. AsfortherestrictionofBtob,thisismoresubtle,butitisnot hardtowritedown:ifX,Yareinb,andZ,generates g,,thenad(X)oad(¥)(Z,)=a(X)a(Y)Z,, 50BUX,Y)=Y.a(X)a(¥), thesumovertheroots;viewingBly ‘asanelement ofthesymmetric square Sym?(h*), wehave 1B=5Ta (1421) $142. About theKilling Form 207 ‘Akeyfactfollowing from this—one that, ifnothing else, makes picturing 'b*with theinner product Binvolve lesseyestrain—is (14.22) Bis positive definite onthereal subspace of¥spanned bythevectors {Heae R). Indeed, allroots takeonrealvalues onthisspace (since all@(H,)€ZcR), s0forHinthisrealsubspaceofb,B(H,H)isnon-negative, andiszeroonlywhen alla(H) =0,which implies H=0,since theroots span b>. ToseethattheKilling form isnondegenerate onallofg, weneed theuseful identity: BEX, ¥},Z) =BX KZ). (1423) forallX,¥,Zing,Ths follows from theidentity Trace((X¥—PX)Z)=Trace(X(¥Z —Z¥) foranyendomorphisms X,¥,Zofavectorspace.Andthis,inturn,follows from Trace(¥XZ —XZY) =Trace(L¥, XZ}) =0. {Animmediate consequence of(1423)isthatifaisanyidealinaLiealgebra§thenitsorthogonal complement a!withrespecttoBisalsoanideal.Inparticular, ifgissimple, thekernel ofBiszero (note that thekernel cannot begsince itdoes notcontain b).Since theKilling form ofa direct sum isthe sum oftheKilling forms ofthefactors, itfollows that theKilling form is nondegenerateanasemisimpleLiealgebrag OneofthereasonstheKillingformhelpstopictureb*isthefactmentioned above: Proposition 14.24. With respect toB,thelinespanned byeach root isperpen- dicular tothe hyperplane Q,, [Asweobserved, thisisequivalent tosaying that theinvolutions W,above aresimply reflections inhyperplanes, and inturn tosaying that thewhole Weyl group isorthogonal. Note also that Proposition 14.24 thereby followsimmediately fromtheFact14.11:fromthedefinitionofBabove,itisclearlyinvariant under anyautomorphism ofg.Nevertheless, wewouldprefernotto relyonthisfact; andanyway giving'a direct proof oftheproposition isnot bard, interms ofthepicture wehave oftheadjoint action ofgonitself.To provetheassertiona1.0,itsuficestoprovethedualassertionthatH1.Hy forallHintheannihilator of.Butnowbyconstruction H,isthecommutator[X,,ZJofanelementX,€9,andanelement¥,€g-,.Using(14.23)wehaveforanyHin b, BOE, H)=BUCXey YJ,H)=BUX, (Yor HD) =BX,,a(H)1.)=aH)BIX,,¥) (1425) which vanishes since a(H) =0 208 14,TheGeneralSet-up:AnalyzingtheStructure Note that asaconsequence ofthis, wecancharacterize theWeyl chamberassociated toanorderingoftherootsasexactlythosevectorsintherealspanoftherootsforminganacuteanglewithallthepositiveroots(or,equivalently,‘withalltheprimitive ones); theWeyl chamber isthusthecone whose faces lie inthehyperplanes perpendicular totheprimitive positive roots. Equation (14.25) leads toaformula fortheisomorphism ofhwith b* determined bytheKilling form. First note that forH=HH,itgives BUH, H,)=2B\Xx, ¥)#0, forifB(X., ¥)were zero wewould have B(H,, H)=Oforall H,contradicting.thenondegeneracy ofBonh.TheelementT,ofwhichcorresponds toa€h*bytheKillingformisbydefinitiontheelementofthatsatisfiesthecondition BUT, H)=a(H) forall Heb, (1426) Looking at(14.25), weseethat T,=H,/B(X., Y_)=2H,/B (HayHy). This proves Corollary 14.27. Theisomorphism ofb*andtydetermined bytheKilling form Bcarries &(0T,=(2/B(Hy, H,))° H,. TheKilling form onh*isdefined byBla,A)=BUT. T). Exercise 14.28. Show that theinverse isomorphism from ftob*takes H,to (2/B(e, «))-a ‘Theorthogonality ofWy,canbeexpressed bytheformula 2B(6,9), Wap) = p—=O sO)=B~ata.a) Comparing with (14.8)this says: Corollary 1429. If«andfare roots, then 2B, 2)/B(e, 2)=(H,) fsaninteger. Bytheabove identification ofhwith h*,(14.22) translates to Corollary 14:30. TheKilling form Bispostive definite ontherealvector space spanned bytheroot lattice Ap. Note that itfollows immediately from (14.22) that theWeyl group 13is Finite, being simultaneously discrete (IBpreserves thesetRofroots ofqand hencethelatticeA;itfollowsthat9Bcanberealizedasasubgroup ofGL,Z) $142, About theKilling Form 209 andcompact (8Bpreserves theKilling form, andhence isasubgroup ofthe orthogonal group O,R.) Alternatively, WBisasubgroup ofthepermutationgroupofthesetofroots. ‘Asweobserved,theKillingformonb*ispreservedbytheWeylgroup.Infact, incase gissimple, theKilling form is,uptoscalars, theunique inner product preserved bytheWeyl group. This willfollow from Proposition 14.31. Thespace b*isanirreducible representation oftheWeyl group BW. Proor. Suppose that 3<b*were preserved bytheaction of4B.This means thatevery roota€b¢ofgwilleitherlicinthesubspace3orbeperpendicular toit,ie,foreverya3andf¢3wewillhaveP(H,)=0.Weclaimthenthat thesubspace4!ofspannedbythesubaigebras {8.}zeywillbeanidealingClearly itwillbeasubalgebra; thespace spanned bythedistinguished sub-algebrass,corresponding tothesetofrootslyinginanysubspaceofh¢willbe.Toseethatitisinfactanideal, let¥€g,beanelement ofarootspace. Then forany ag,wehave UZ] emp=0 since a+flisneitherin3norperpendicular toit,andsocannotbearoot;and C,H.) =—(H,, Y]=f(H,):¥=0. Thus, ad(Y) kills g;since, ofcourse, allof1itself willpreserve git follows thatgisanideal. Thus, either alltheroots liein3andsoj=h*,orallroots areperpendicular togandcorrespondingly 3=(0) a NotethatgivenFact14.11,wecanalsoexpressthelaststaiementbysayingthat(incase gissimple) theKilling form onbistheunique form preserved byevery automorphism oftheLiealgebra gcarrying lytoitself. Aswewill sce,inpractice thisismost often how wewillfirstdescribe theKilling form. Exercise 14.32. Find theKilling form ontheLiealgebras s1,€andsly€by ‘explicit computation, andverify thestatements made above inthese cases. Exercise 14.33°. Ifasemisimple Liealgebra isadirect sum ofsimple sub- algebras, then itsKilling form istheorthogonal sum oftheKilling forms of thefactors. Show that, conversely, ifthe roots ofa semisimple Liealgebra lie inacollection ofmutually perpendicular subspaces, then theLiealgebra decomposes accordingly. Exercise 14.34*, Suppose gisLiealgebra that hasanabelian subalgebrabysuchthatghasadecomposition (14.1),satisfyingtheconditions ofFacts142and 146. Show that gissemisimple, andhisaCartan subalgebra. 210 14,The General Setup: Analyzing theStructure ‘The preceding exercise canbeused instead ofWeyl’s unitary trick orany abstract theory toverify thatthealgebras wemeetin thenext fewlectures are allsemisimple. Itistempting tocallsuch aLiealgebra “visibly semisimple.”‘ThediscussionofthegeometryoftherootsofasemisimpleLiealgebrawill becontinued inLecture 21andcompleted inAppendix D.The Killing form becomes particularly useful inthegeneral theory; forexample, solvability andsemisimplicity canbothbecharacterized bypropertiesoftheKillingform(seeAppendix C). Exercise14.35¢.Showthatb=f®@e>oGeismaximalsolvablesubalgebraofg;biscalledaBorelsubalgebra. Showthat€),.»o9.isamaximal nilpotentsubalgebra ofg.ThesewillbediscussedinLecture25. Exercise 14.36%, Show that theKilling form ontheLiealgebra glyisgiven by the formula BUX, ¥)=2mTHX o¥)—2TH(X) TH(Y). Findsimilarformulasforsl,$0q,aridsPq,showingineachcasethatB(X,¥)isaconstant multiple ofTH(X oY). Exercise 14.37. IfGis@realLiegroup, theKilling form onitsLiealgebra 9=T.Gmay notbepositive definite, When its, itdetermines, bylettrans- lation, aRiemannian metric onG.Show that theKilling form ispositive definite for G=SO,R, butnotforSLR. LECTURE 15 sl,C€ and sl,C Inthislecture, wewillillustrate thegeneral paradigm oftheprevious lecture byapplyingtotheLiealgebrasl;thsitypicaloftheanalysesofspecificLiealgebras carried outinthis Part. Westart in§15.1 bydescribing theCartan subalgebra, roo, root spaces, te,foral,€ ingeneral. Wethen givein§152detailedaccountof ‘therepresentations ofa,whichgeneralizesdiectyto},C;in particular,wededuce theexistencepartofTheorem14.18for#€.In§15.3wegiveanexplicitconstruction oftheireduciblerepreseatations ofsf,C‘using theWeyl construction introdvced inLecture 6;analogous constructions ofthe itreducible cepresentationsoftheremainingclassicalLiealgebraswillbegivenin§17.3 and§19,5. This section presupposes familiarity with Lecture 6and Appendix A,but canbeskipped bythosewillingtoforego§17.3and19.5aswell.Section 15.4requires:‘essentially thesamedegreeofknowledge ofclassical algebraic geometry as§§11.3and1344(itdoes notpresuppose §15.3), butcanalso beskipped. Finally, §15.5 describes representations ofGLC;thisappearstoinvolvetheWeylconstruction butinfactthe toainstatement,Proposition 1547(andevenitsproof)canbeunderstoodwithoutthepreceding two sections $151: Analyzing lye §152: Representationsofs,Cands1,¢ 5153: Weyl construction andtensor products 4154: Some more geometry $185: Representations ofGLC $15.1. Analyzing s{,C Tobegin with, wehave tolocate aCartan subalgebra, andthisisnothard;asinthecaseofal,andsly€thesubalgebra ofdiagonalmatriceswillworkfine. Writing H,forthediagonal matrix E,,,that takes e,toitself andkills e, 2 15,sl4Candsf, forj#i,wehave =ayHly+aglly+o+aly:a,+034°+y=0}; ‘note that His notinb.Weeancorrespondingly write be=C{LyyLayevesLyh(Ly+Lato+Ly=O) where Li(H,) =6,,.Weoften write L,fortheimage ofL,inb* Wehave already seen how thediagonal matrices actonthespace ofalltracelessmatrices:ifEistheendomorphism of€*carrying¢,toe,andkillingeyforall k#j,then wehave ad(a,Hy+a,H,+>+a,H,)(E,,) =(a,~a)Exjs(15.1)or,inotherwords,E,18aneigenvector fortheactionofbwitheigenvalueLy~Lin particular, therootsofsl,CarejustthepairwisedifferencesoftheL;. Before wetrytovisualize anything taking place inbor*,letustake @ ‘moment outanddescribe theKilling form. Tothisend, note that theauto-morphism »ofC*sending e,toe, ¢;t0 ~e,andfixingeforallk#1,jinduces anautomorphism Ad(p) oftheLiealgebra s1,C (oreven gl,(C)) that carries {toitself, exchanges H,andH,andfixes alltheother Hy.Since theKilling form onhmust beinvariant under allthese automorphisms, itmust satisfy BUL, Ly)=BlLy, Ly)foralland jandB(L, U4)=B(Ly La)forall, jand#1,{itfollowsthatonbitmustbealinearcombination oftheforms BLaH, EHH) =Lab and BU, HE bil) =Dayiby Onthespace {Ja): Ya,=0}, however, wehave 0=(Ya)(Lb) = Sayb, +Yay, 80infactthese twoforms aredependent; andhence wecan ‘write theKilling form simply asamultiple ofB’.Similarly, theKilling formconb*mustbealinearcombination oftheformsB'(S:aby,Shula)=5.andB'SaL,LbiLa)=Lyoiabj; thecondition thatB(Y,aL,),bL,)=0whenever a,=43="°*= 6,0FBy=by=": =b, implies that itmust bea ltiple of =! 1 BQaeDba)=+)Sab7Eat j (152) Lab5Bab, Wemay,ofcourse,alsocalculatetheKillingformdirectlyfromthedefini-tion, By(14.21, since theroots ofslxCare{Lj~Lj}iyy wehave BYaie, Yb) =Laos(= Nb ~6) =LiLaerlauds+ayby—aby—ab). NotingthatJ°,,,4=—o,and,similarly, "41h=—h,thissimplifies to S154. Analyzing 1,6 213 BUYHyYbth)=2mYay (153) Itfollows with alittle calculation that thedual form onh*is BYabe, Zbl)=(1/2 Epab~ Tisai. (15.4) Itisprobably simplerjusttothinkofthisastheform,uniqueuptoscalars,invariant underthesymmetric group&,ofpermutations of(1,2,...,2}.TheL,,therefore, allhave thesame length, andtheangles between allpairs are thesame.Topicturetherootsinh*,then,weshouldthinkofthepointsL,assituated atthe vertices ofaregular(n—1)-simplex A,withtheoriginlocated atthebarycenter ofthat simplex. This picture iseasiest tovisualize inthe special case n=4,where theL,willbelocated atevery other vertex ofaunit cube centered attheorigin: by ly if i H(53) oe i Now, aswesaid, theroots ofal,C arenow just thepairwise differences of theL;.The root lattice Aythey generate canthus bedescribed as Ag=(DadeaeZ,La,=OELy=0) Boththerootsandtherootlatticecanbedrawninthecaseofs1,C:ifwethinkofthevectors L,€h*asfour ofthevertices ofacubecenteredattheorigin, therootswillcomprise allthemidpointsoftheedgesofasecondcubewhose linear dimensions are twice the dimensions ofthe ist byob Lucky \ i + fuels (156) 1 Lyk | i ae”~ an ky m4 15, eCard aie ‘The next step, finding thedistinguished subalgebras s,,isalso very easy. ‘Therootspace g,,-1, corresponding totherootLy~L,isgenerated byEj. s0thesubalgebra 5,1, igenerated by Buy Bsn and (Ey Eyal =Hh—Hy ‘Theeigenvalue ofH,~H,actingonE;,is(L,—L,)(U,—H,)=2,sothatthe correspondingdistinguished elementH,,-.,inhmustbejustH,—H,.The annihilator,ofcourse,isthehyperplaneQ),-,,={La,Ly:a=aj};notethat thisisindeed perpendicular totheroot L;~ L,with respect totheKilling form Basdescribed above Knowing theH,weknow theweight lattice: inorder foralinear functional Ya,L;€h* tohave integral values onallthedistinguished elements, itis clearly necessary andsufficient that allthea,becongruent tooneanother moduloZ.Since¥:L;=0inh,thismeansthattheweightlatticeisgivenas A=2{LyyooLaAELy=O Insum, then, theweight lattice ofs|,€ mayberealizedasthelatticegenerated bythevertices ofaregular (n—\)-simplex Acentered attheorigin; andthe rootsasthepairwisedifferencesofthesevertices.‘While weareatit,having determined AgandAwwemight aswellcompute thequotient Aw/A.,. This ispretty easy: since thelattice Awcanbegenerated bbyAxtogether withanyoftheverticesL,ofour simplex, thequotient Aw/Ax. willbecyclic,generated byanyL,;since,modulo Ag, 0=Sy(ly— Ly)=my~LyLy=by. weseethatL,hasorder dividing ninAw/A,. Exercise 15.7. Show that L,hasorder exactly ninAw/Ag, 80that Aw/Ag = Zt. FromtheabovewecanalsosaywhattheWeylgroupis:thereflection inthehyperplane perpendicular totherootL,~L,willexchange L,andL,€h* andleave theother Lyalone, sothat theWeyl group 8Bisjust thegroup ©, ‘acting asthesymmetric group onthegenerators L,ofh*.Note that wehave already verified thatthese automorphisms ofh* docome from automorphisms ofthe whole Liealgebra s1,C preserving b Tocontinue, letuschooseadirection, anddescribe thecorresponding Weylchamber. Wecanwriteourlinearfunctional |as Wat) =Dem, with Te;=0;letussuppose thatcy>¢,>--"> cy.Thecorresponding ordering oftherootswillthenbe Rt={L,-Lzi< J} and S151, Anang Le as . R= {L,-LyJ<I} “The primitive negative roots forthisordering aresimply theroots Li. ~Ly. (Note that theordering othe oot depends only onthe reaie ie ofthe ¢,,80 thattheWeyl group actssimply transitively onthesetoforderings.) The (Closed) Weyl chamber associated tothisordering willthen betheset Ya Qatia 2e2 2a) (newaytodescribthigeometely iststhatwetakethebrgcentiesubdivision ofthefacesofthesimplex A,theWeylchamber willbetheconeoverone(n—2}-simplex ofthebarycentric subdivision: eg,inthecasem=4 SZ Itmaybeeasiertovisualizethecasen=4ifweintroducetheassociated cubes:intermsofthecubewithverticesatthepoints:+L,,wecandrawtheWeylChamber at | ~ i 4 (158) \ —_ite 216 15,alC and si, Alternatively, interms oftheslightly larger cube with vertices atthepoints £2Lq weean draw Was byt \ i4 (159) 11,158 LoessweST o— From the fistofthesepictureswesethattheedgesoftheWeylchamberare theraysgeneratedbythevectorsLy,Ly+LayandLy+L,+Ly;andthat thefaces oftheWeyl chamber aretheplanes orthogonal totheprimitive negative roots Ly~Ly,Ly~La, and Ly~ Ly.The picture ingeneral is analogous: fors1,C, theWeyl chamber will bethecone over an(it~ 2} simplex, with edges generated bythevectors LysLatLayytbatbapeeeglyttbt=hy The faces of#”willthus bethehyperplanes Dit, =(Lalya,=ays} perpendicular totheprimitive negative roots Lys —Lie ‘Note theimportant phenomenon: theintersection oftheclosed Weyl chamber with thelattice Awwillbeafreesemigroup N*™* generated bythefundamental weights,=Ly+:+L,occurting alongtheedgesoftheWeylchamber. One aspect ofitssignificance that isimmediate istha itallows us toindex theirreducible representations s1,C nicely: foranarbitrary (n~Ip tupleofnaturalnumbers(4),...,dy-1)€ N°wewilldenotebYTy...theirreducible representation ofsi,C with highest weight a,L, +.a,(Ly +L3)+ Pe eyLyHtbag)(ay+oHaag +go yada+ ‘Thisalsothas theniceconsequence thatoncewehavelocatedtheireducible representations ¥"withhighestweightLy+"+Ly,thegeneralirreducible 152. Representation ofs,C andsi,C 2 representation T,,,...,, withhighest weight Yla(L, +--+ L))willoccur inside thetensor product ofsymmetric powers Sym*'¥"@Symer¥1@---@ Symresvirt ofthese representations. Thus, theexistence part ofthebasic Theorem 14.18 jisreduced tofinding thebasic representations V; wewilldothisindue course, though atthispoint itisprobably nottoohard anexercise toguess what they are. §15.2. Representations ofst,€ and sl,C Webeginasusualwiththestandard representation ofsi,ConV=C*.Thestandard basis vectors e,ofC*areeigenvectors fortheaction ofb,with eigenvalues L;,sothattheweight diagram looks like by\// y, VY «with thereference cube drawn aswel, J ‘Thedualrepresentation V*ofcoursehasweights—L,corresponding tothe vectors ofthedual basis e?forV*,sothat theweight diagram, with its reference cube, looks like LGaA“ Note that thehighest weight forthisrepresentation is—_L4, which liesalong.thebottomedgeoftheWeyl chamber,asdepictedinDiagram(15.8).Notealso thattheweights oftherepresentation A'V—the triple sums Ly+L;+Ls, Ly+Ly+LayLy+Ly+Lasand Ly+Ly+L,ofdistinct weightsofV—are This suggests that welook next atthesecond exterior power A?V. This is asix-dimensional representation, with weights L,+L,thepairwise sums of distinct weights ofV;itsweight diagram, initsreference cube, looks like pay aa G ‘Thediagramshowsclearlythat\*Visirreducible sinceitisnotthenontrivialunion oftwo configurations invariant under theWeyl group S,(and all weights occur with multiplicity 1).Note also that theweights aresymmetric about theorigin, reflecting theisomorphism ofA?V with (A?V)* =A%(V*). Note that thehighest weight L,+L, oftherepresentation AV isthe primitive vector along thefront edge oftheWeyl chamber #*aspictured in Diagram (15.8). Now, wehave already seen that theintersection oftheclosed $152. Representations ofsand1, 219 Weyl chamber with theweight lattice isafreesemigroup generated bytheprimitivevectorsalongthethreeedgesof#”—thatieveryvectorin#°7Ay isanon-negative integral linear combination ofthe three vectors L,Ly +Lay and Ly+L, +Ly.Asweremarked attheendofthefirst section ofthis lecture,itfollowsthatwehavepravedtheexistencehalfofthegeneralexistence anduniqueness theorem (14.18)inthecase oftheLiealgebra st,C. Explicitly, since ¥,AY, and AY =V*have highest weight vectors with weights Ly, Ly+Ly,andLy+Ly+Ly,respectively, itfollows that therepresentation Sym"V@Sym*(\?V)@ Sym'(\°V) contains ahighest weight vector with weight aly +b(Ly +Ly)+{Ly+Ly+Ly),andhenceacopyoftheirreducible representation Ty,Withthishighest weight. Letuscontinue ourexamination ofrepresentations ofgC with apair of tensor products ofthethree basic representations: ¥@A*V andV@AV. {Asforthefirstofthese, itsweights areeasy tofind: they consist ofthesums2a+L,(whichoccuronce,a8thesumofL,andLy+L,)andLy+Ly+Ly(which occur three times). The diagram ofthese weights looks like WS) (We have drawn only thevertices oftheconvex hull ofthisdiagram, thus omitting theweights L,+L, +La;they arelocated atthecenters ofthe hexagonal faces ofthis polyhedron) Now, therepresentation V@A'V cannot beirreducible, foratleast a couple ofreasons. First off,just bylooking atweights, weseethat the irreducible representation W=T),.,owith highest weight 2L,+L2.can havemuhiplicity atmost2ontheweightL,+Ly+Ls:byObservation 14.16,theweightspaceM%,.1,+1,i8generatedbytheimagesofthehighestweightvector beWatson, bysuccessive applications oftheprimitive negative root spaces Sry-tarBryty»AGGy,-r4-ButLy+Ly+Lyisuniquelyexpressibleasasum Of2Ly +Lyandtheprimitive negative roots: Ly+Ly+by=Wy$batUlaLa)+(Ly—Lak 20that Vissegeny 18generated bythesubspaces gr,-4(G1,-14(0)) and Si-ts(Oey-1,(0}). Wecaninfactcheck thattherepresentation Ty, takes ontheweight L,+L, +Lywith multiplicity 2bywriting ovtthese have 84-14 (85-10) =C-Ey,(Es,2(€,ler A€2))) =C-E,,,(e, @(e, A€3)) =C-(e, @le,aes)+e,Ole,A€3))- ‘ThisisinfactwhatiscalledforinExercise 15.10. ‘Alternatively, forgetting weights entirely, wecanseefrom standard multi- linearalgebra thattherepresentation V@A?Vcannotbeirreducible: wehave @VONVaNY whichisobviously surjective. Thekernelofthismapisarepresentation withthesamesetofweightsasV@AV(buttakingontheweightsLy+Lj+Ly tation F),,0 with highest weight 2L,+Ly. Exercise 15.10,Provethatthekernelof¢isindeedtheirreducible represen- tation Fy1.9 Finally, consider the tensor product V@/%V. This has weights2L,+Ly+Ly=Ly—Lyeachoccurring once,and0,occurring fourtimes.Its weight diagrams thus look like \OK ‘This wemay recognize assimply adirect sum oftheadjoint representation with acopy ofthetrivial; thiscorresponds tothekernel and image ofthe obvious contraction (ortrace) map §152. Representations of#,€ands1,C 2 V@AV =V@V*=Hom(y, ¥)-+€. (Note that theadjoint representation istheirreducible representation with highest weight 2L,+Ly+Ly,orinother words therepresentation F,.,1.) Exercise15.11.Describetheweightsoftherepresentations Sym*V,anddeducethat they areallirreducible. Exercise 15.12. Describe theweights oftherepresentations Sym"(\?V), and deduce thatthey arenotirreducible. Describe maps 94:Symm"(MPV)+Sym™-3(V) andshow that thekernel ofg,istheirreducible representation with highest weight n(L, +L3) Exercise 15.13. The irreducible representation I... with highest weight 3Ly+2L;+L occurs asasubrepresentation ‘ofthe tensor productV@AV@NVlyinginthekernelofeachofthethreemaps, VONV@NV=AV@NVY VONV@NV>NVONV &AV V@NVONV =VONV*@VY+VONVtXVOV obtained bywedging twoofthethree factors Isitequal totheintersection of these kernels? Totestyour graphic abilities, draw adiagram oftheweights (ignoring multiplicities) ofthisrepresentation. Representations ofs1,C ‘Oncethecaseofst,Cisdigested,thecaseofthespeciallineargroupingeneral offersnosurprises;themaindifferenceinthegeneralcaseisjusttheabsence ofpictures, Ofcourse, thestandard representation VofslyC hashighest weight Ly,andsimilarly theexterior power /*Visirreducible with highest weight Ly+--+ LyItfollows thattheirreducible representation Ty... ,with highest weight (a,+--+ a,-:)Ly +" +ay-4Ly-1 will appear inside the tensor product Sym"V @Sym"(A7V) @--- @Sym'(N"V), demonstrating theexistence theorem (14.18) forrepresentations ofs1,C. Exercise 15.14. Verify thattheexterior powers ofthe standard representationsofsi,Careindeedirreducible thoughthisisnotnecessaryforthetruthofthe lastSentence). 22 15,a©and sl, §15.3. Weyl’s Construction and Tensor Products Attheendofthepreceding section, wesawthattheirreducible representationFa.osq-«OFUCwithhighestweight(ay+2°*+ay-y)by °°"+ytLq-Will‘appear asasubspace ofthetensor product Sym" V@Sym(3P)@ ~~@Sym*-H(V), ‘orequivalently asasubspace ofthedthtensor power V®ofthestandard representation ¥.Thenatural question is,how canwedescribe thissubspace? We have seen the answerinonecasealready(twocases,ifyoucountthetrivial ‘answerFr,=Sym*Vinthecasen=2):therepresentation I,ofslyCcanbe realized asthekernel ofthecontraction map Sym*V @Sym*(A?V) -+Sym*""V @Sym?""(7). This raises thequestion ofwhether therepresentation I,caningeneral be described asasubspace ofthetensor power @(Sym*(\'V)) byintersecting kernels ofsuch contraction/wedge product maps. Specifically, foriandjwith i+j-SmwecandefinemapsSym"V@Sym™(A7V) @++@Sym"(AV) =AV@NV®Sym"V @--@Sym*"(NV)@ ~~ @Sym") @-@Sym*4A1V) andwehayesimilarmapsfor{<jwith+j=mandievenwith2i>n;there arelikewise analogously defined maps inwhich wesplit offthree ormorefactors.Therepresentation T,,.._.,_, isinthekernelofallsuchmaps;andwemayaskwhethertheintersection ofallsuchkernelsisequaltoI.Theanswer, itturns out,isno.Itis aworthwhile exercise tofindanexample ‘ofarepresentation I,thatcannot berealized inthisway.) There is,however, another wayofdescribing IP’,asasubspace ofV®: infact, wehave already ‘metthese representations inLecture 6,under theguise ofSchur functors ot Weyl modules. Infact, attheend ofthis lecture wewill seehow todescribe themexplicitly assubspaces oftheabovespaces@(Sym*(A'V)). Recallthat forV=€*anmdimensional vector space, and any partition RAZA ezBAO, wecan apply theSchur functor S,toVtoobtain arepresentationS,V=S,(C*)ofGL(V)=GL,(C).I'd=YA,thiswasrealizedas SV=Vc,=VBes,Var wherec,istheYoungsymmetrizer corresponding toA,andV,istheirreducible representation ofS,corresponding to2. ‘Wesaw inLecture 6that S,Visanirreducible representation ofGL,C. It follows immediately thatS,Vremains irreducible asarepresentation ofSL,C, $153, Wey! Construction andTensor Products mm since any element ofGL,C isascalar multiple ofanelement ofSLC. Inparticular, itdetermines anirreducible representation oftheLiealgebrasl,C: Proposition 15.15. Therepresentation S,(C") istheirreducible representation ofsI,C with highest weight yLy +AgLy +°* +AyLy. Inparticular, $,(C*) andS,(C*)areisomorphicrepresentations ofsi,Cif andonly ifA,—jyisconstant, independent ofi.Torelate thistoourearlier notation,wemaysaythattheirreduciblerepresentation T,,...,,of{4Cwith highestweighta,L,+a,(Ly+Ly)+°"+a,.,(Ly+*-+Ly-,)isobtained bbyapplying theSchurfunctorS,tothestandard representation V,where ema EH ats BOEOyetysonedents) (IfwewantauniqueSchurfunctorforeachrepresentation, wecanrestrictto those 2with 2,=0.)Interms oftheYoung diagram forA,thecoefficients a,=A,—Ayy arethedifferences oflengths ofrows. Forexample, ifn=6, aay =a as « istheYoung diagram corresponding toT,,0,3,1+ PROOFOFTHEPROPOSITION. InTheorem 6.3wecalculated thatthetraceofa diagonal matrix with entries x;,...,x,on$,(C*) istheSchur polynomial 'S,(0y, -..%). ByEquation (A.19), when theSchur polynomial iswritten out ittakes the form Slee) =Ma+LKayMy (15.16) where M,isthesum ofthemonomial X*=xftxJ?+...-xf* andalldistinct monomials obtained from itbypermuting thevariables, andtheK,,are certain non-negative integers called Kostka numbers. When S,(C") is diagonalized withrespect tothegroup ofdiagonal matrices inGL,(C), it isalso diagonalized with respect toh<sl,(C). There isonemonomial in thedisplayed equation foreach one-dimensional eigenspace. The weights ofS,(C")asarepresentation ofsl,(C)therefore consistofall WyLy+aby bo +tab each occurring asoften asitdoes inthemonomial X*inthepolynomial 24 15,al,C and o€ S,G¢4y---5%4) Since thesum isover those partitions jforwhich thefirst nonzero A,~jispositive, thehighest weight thatappears isAyLy+AL, + +AyLy, which concludes theproof. [Infactonecandescribe anexplicitbasisofeigenvectors forS,(C*) which correspond tothemonomials that appear in(15.16), ef.Problem 6.15 orProposition 15.55. fa} Inparticular, wehave (byTheorem 6.3)formulas forthedimension ofthe representation with given highest weight. Explicitly, oneformula says that i (toto )tii a Peta tirt asin Conmted™Dey j-t ‘Aswesaw intheproof, this proposition also gives themultiplicities ofall ‘weight spaces astheintegers K,,thatappear in(15.16), which have asimple ‘combinatorial description (p.456): thedimension oftheweight space with weight 1intherepresentation S,(C*)isthenumberofwaysonecanfillthe Young diagram ofAwith jyV's,ty2',., M8,insuch away that theentries ineach rowarenondecreasing andthose ineach colunn arestrictly increasing. Exercise 15.18. Use theformula incase n=4tocalculate thedimensions of theirreducible representations F,s,9andf),s,,ofslC.Intheformercase,usethistoredo Exercise 15.10; inthelatter case, todoExercise 15.13. Exercise15.19%,Usethisformulatoshowthatthedimension oftheirreduciblerepresentation T,, ofsly with highest weight aly+(Ly +La) is (a+b +1)(a +1)(b +1/2. This isthesame asthedimension ofthekernel ofthecontraction map tau:Symm*V@Sym"V*-»Sym?“!V@Sym1V*, Usethistogiveanother proof oftheassertion madeinClaim 13.4that Fi this kernel. Exercise 15.20*. Asanapplication oftheabove formula, show that ifVisthe standard representation ofs,C, then thekernel ofthewedge product map VONV+ANY isthe irreducible representation Ty,o,..0,1,0,.. with highest weight 2Ly+Ly+" +Ly:andthattheirreducible representation Ty.»,1,0,.. with hhighest weight k-Ly+Lyisthekernel oftheproduet map V.@Sym*V—+Sym", Exercise 15.21*. Show that theonly nontrivial irreducible representations ofsI,Cofdimension lessthanorequaltonareVandV*. ‘One important consequence ofthefactthattheirreducible representationsofsi,CareobtainedbyapplyingSchurfunctorstothestandardrepresentation §153. Weyts Construction and Tensor Products ns isthat identities among theSchur-Weyl functors giverisetoidentities among representations ofGL,(and hence SL,andsi,),aswesawinLecture 6.For ‘example therepresentation Sym4(V) @Sym’(V) @--@Sym*(V) (15.22) isadirect sum ofrepresentations §,(V)®@),K,S,(V),whereK,,isthe coefficient described above, The particular application ofthisprinciple that ‘wewillusemost frequently inthesequel, however, istheconsequence thatoneowsthedecomposition oftensorproductofanytwoirreduciblerepresen- ationsofs},€:specifically, thetensorpowerS,(V)®S,(V)decomposes intoadirectsumofirreducible representations SOS) =ONS) (1523) where thecoefficientsN;,,aregivenbytheLittlewood-Richardson rule,which isaformulaintermsofthenumberofwaystofilltheYoungdiagrambetween Zand vwith pry1s,ty2'S,...yMyms,Satislying acertain combinatorial condition described in(A.8). Exercise 15.24. Use the Littlewood-Richardson rule toshow that the representation T,,44,,..e.¥%., OCCUFS exactly once inthetensor product Faces®Toot Aspecial case ofthisistheanalogue ofPieri's formula, which allows us todecompose thetensor product ofanarbitrary irreducible representation with either Sym'V =Tj,o,.6 orthefundamental representation AV= Tp,...1.0,..,0e (where theToccurs inthekthplace): Proposition 15.25. (i)Thetensor product ofTy,,...0,., withSym'V =Tj,o,..0 decomposes into adirect sum: Fasetet ©Fn =Dlijoahert thesumoverallbysyes)frwhichtherearenon-negative integers¢ys--s¢ywhasesumisk,Withey,tyforSiS~1,andwithby=a,Gaot for\ sisn—. i)Thetensor product ofT,...0, WithNV =To,_..0,1,0,...0 decomposes into adireet sum: Fact OF 01.0-.0% Dliutee ‘thesum over all(by, ...,b,-1) forwhich there isasubset Sof{1,....n} of cardinality k,suchthatifi¢Sandi+leS,thena,>0,with a—1 ifi¢Sandities he fat ilieSandi+ ies a, otherwise. 226 15.slyCandsi, Proor. This issimply amatter oftranslating theprescriptions of(68) and(69),whichdescribethedecompositions intermsofaddingboxestotheYoungdiagrams. In(the care thenumber ofboxes added totheithrow, andin{iiSisthesetofrowstowhichaboxisadded. a Exercise15.26.Verifythedescriptions inSection2ofthislectureofV@?VandV@A'Y,whereVisthestandardrepresentation ofs1,C. Exercise15.27.UsePierisformula(withn=4)twicetofindthedecompositioninto irreducibles ofV@A?@AV,whereVisthestandardrepresentation ofslC. Use this toredo Exercise 15.13. Exercise 15.28, Use Pier’ formula toprove (135). You may also want tolook around inLecture 13toseewhich other ofthedecompositions found there byhand may bededuced from these formulas. Exercise 15.29. Verily that thestatement ofExercise 15.20 follows directly from Pier’s formula. Inthefollowingexercises,V=C*isthestandardrepresentation ofsi,€. Exercise 15.30. Consider nowtensor products oftheform A'V@A'V,with, say, k>l.Show that there isanatural map RV @AV NOV QNY given bycontraction with theelement “trace” (or“identity") inV@ V*= End(V). Explicitly, thismap may begiven by (A ABln AoA mH) ap(DOA AOAMIBOW,AABA AHe What istheimage ofthis map? Show that theKernel istheirreducible representation To,..o.,0,..0,1,0... With highest weight 2L,++--+2L,+ Lys tot by Exercise 15.31*.Carry outananalysis similar tothatofthe preceding exercise forthemaps Sym*V@Sym'V-»Sym**!V@Sym!"!V defined analogously. Exercise 15.32*. Asaspecial case ofPieris formula, weseethat ifVisthe standard representation ofs,C, thetensor product $154, Some More Geometry n NVORV=DSa,..2tet0M) =DMo.1.0..0 0.40.08 where intheithfactor the1'soccur inthe(k—ithand(k+ithplaces. A:thesametime,ofcourse,weknowthat AV @AV=Sym(ANV)@ AAV). Ifwedenotetheithtermontheright-hand sideofthefirstdisplayedequationforMV@MYby@,,showthat Sym(MV)= GO, and AMV) =@ Ox. Exercise 1533. Asanother special case ofPier's formula, weseethat the tensor product Sym*V@Sym'V=@Susis-ol¥) =Bhrco.0 ‘Atthesame time, ofcourse, weknow that Sym*V@Sym*V=Sym?(Sym*V) @A*(Sym*¥). Whichofthefactorsappearinginthefirstdecomposition lieinSym?(Sym*V),andwhich inA%(Sym*V)2 ICfollows from theLittlewood-Richardson rule that if,,andvallhave Atmost tworows, then thecoefficient Ny, iszetoorone(and itiseasy to saywhich occurs) Inparticular, fortheLiealgebras sl,C andslyC, the decomposition ofthetensor product oftwo irreducible representations is always multiplicity free. Groups whose representations have thisproperty, such asSU(2), SUG), andSOG) which aresoimportant inphysics, arecalled “simply reducible,” cf.[Mack]. §15.4. Some More Geometry LetVbeann-dimensional vectorspace,andG(k,n)=Gtk,V)=Grass,theGrassmannian ofk-planesinV.Grass,Visembedded asasubvariely oftheprojective space P('*V) bythePlicker embedding: p:Grass,¥+PUNY) sending theplane Wspanned byvectors 0,..,tyt0thealternating tensorbyA-°*4Oy.Equivalently, notingthatifW<Visak-dimensional subspace, thenA'Wisalinein(RY,wemaywritethissimplyas psWW, ns 15.lC andoh ‘Thisembedding iscompatible withtheactionofthe general linear group: PSL,C =Aut(P(V)) ={0€Aut(PUMY)): o(G(k, V))=Gk, ¥)}”. This follows from afactinalgebraic geometry ({Ha]): allautomorphisms‘oftheGrassmannian areinducedbyautomorphisms ofV,unlessn=2k,inwhich case wecan choose anarbitrary isomorphism ofVwith V* andcompose these with theautomorphism that takes Wto(C*/W)*. Here thesuperscript *denotes theconnected component oftheidentity. Asin previous lectures, ifwewant symmetric powers tocorrespond tohomo- geneous polynomials onprojective space, weshould consider thedual situa- tion: G=Grass*V istheGrassmannian ofk-dimensional quotient spaces of V,andthePliicker embedding embeds Gintheprojective space P(M'V*) of one-dimensional quotients ofA'V.‘Thespace ofallhomogeneous polynomials ofdegree monP(A'V*) is naturally thesymmetric power Sym"(A'V). Let(Gq denote thesubspace ofthosepolynomials ofdegreemonP(\*V*) thatvanishonG.EachI(G),is@ representation ofs1,C: 0-+1(G)q.-» Sym™(MV) —+Wy»0, where W,denotes therestrictions toGofthepolynomials ofdegree mon theambient space P(A*V*). Weshall seelater that W,,istheirreducible representation Tp,...o.n.o,... With highest weight m(L, +">+Ly)(thecase ‘m=2willbedealt with below). Inthefollowing discussion, weconsider theproblemofdescribing thequadratic part1(G),ofthe ideal asarepresentation ofsi,C, Exercise 15.34. Consider the first case ofaGrassmannian that isnot a projective space, that is,k=2.The ideal oftheGrassmannian G(2, V)of Zplanes inavector spaceiseasytodescribe:atensor@€\*Visdecomposable ifandonly ifp«p=0(equivalently, ifwethink ofgasgiven byaskew- symmetric xnmatrix,ifandonlyifthePfalfiansofsymmetric 4x4minors allvanish} and indeed thequadratie relations wegetinthis way generate the ideal oftheGrassmannian. We, thus, have anisomorphism 1G), =MV and correspondingly adecomposition into irreducibles Sym7(A7V) &MV®F,2,0....09 where To,2.0,...0 8,a8above, theirreducible representation with highest weight 2(2, Ly)ef.Exercise1532. Exercise 15.38. When k=2andn=4, Gisaquadric hypersurface inPS,so polynomials vanishing onGaresimply those divisible thequadratic poly- nomial that defines G.Deduce anisomorphism. 1G)q =Sym"). 8154. Some More Geometry 29 The firstcase ofaGrassmannian that isnotaprojective space oroftheformG(2,V)is,ofcourse,G(3,6),andthisyieldsaninteresting example. Exercise 15.36. Let Vbesixdimensional. Byexamining weights, show that thespace 1(G), ofquadratic polynomials vanishing ontheGrassmannian GQ, ¥)<PUY) isisomorphic totheadjoint representation ofsigC, ic, that wehave amap g:Sym4(M'V)-» V@V" with image thespace oftraceless matrices. Exercise 15.37. Find explicitly themap gofthepreceding exercise. Exercise 15.38. Again, letVbesixdimensional. Show that therepresentation Sym*(A*¥V) hasatrivial direct summand, corresponding tothehypersurface inP(\?V*) dual totheGrassmannian G=G(3, ¥).< PAY). Ingeneral, theideal 1(G) =G)1(Gy isgenerated bythefamous Pliicker‘equations.Thesearehomogeneous polynomials ofdegreetwo,andmaybewrittendownexplicitly,of,(15.53),(H-P],or[Ha].Inthefollowingexercises,‘Wewillgive amore intrinsic description ofthese relations, which willallow ustoidentify thespace 1(G), they span asarepresentation onsI,C (and to seethegeneral pattern ofwhich theabove arespecial cases). Exercise15.39.ForagiventensorA«AY,weintroducetwoassociatedsubspaces: We(veVionh=}CV and We=(ot EVEUtAAT =O} Ct, where, abusing notation slightly, A*isthetensor Aviewed asanelement of NV =NAVE, Show that thedimensions ofWand W* are atmost kand ‘n~k,respectively, andthatAisdecomposable ifandonlyifWhasdimensionkor W* has dimensionn—k;anddeducethatAisdecomposable ifandonly ifthe annihilator W'ofW*isequal toW. Exercise 15.40. Now letBAM#V* =Av 'V,Wedge product gives amap iNVNOV =ve Using thepreceding exercise, show that Aisdecomposable ifandonly if n(A) AA=06K1V forallZeAM* 230 15,at,Candst, Exercise 1541. Observe that inthepreceding exercise weconstruct amap NOV" ®Sym(NV) AY, or,byduality, amap NOY*@NAVE 4Sym4Nvry (15.42) whoseimageisavectorspaceofquadrics onP(N'V) whose common zeros areexactly thelocus ofdecomposable vectors, that is,theGrassmannian G(k, V).Show that thisimage isexactly thespan ofthePliicker relations above. Exercise 1543. Show that themap(15.42) ofthe preceding exercise isjustthedualofthe map constructed inExercise 15:30, with k=land restricted tothe symmetric product. Combining thiswith theresult ofExercise 15.32 (and assuming thestatement that thePlicker relations doindeed span 1(G),) deduce that intermsofthedescription Sym(AV) =®Ox ofthesymmetric square ofA'Y,wehave Wy=Qo=To,...0.2.0 (theireducible representation with highest weight 2(L, +++ L,))and HO); = DO Hard Exercise 15.44, Show thatinthelastequation thesub-direct sum = @exn isjust thequadratic part oftheideal oftherestricted chordal variety ofthe ‘Grassmannian: thatis,theunion ofthechords LMjoining pairs ofpoints inGcorresponding topairs ofplanes Land Mmeeting inasubspace of dimension atleastk~21+1.(Question:Whatistheactualzerolocusofthese ‘quadrics?) Exercise 1548. Carry out ananalysis similar totheabove torelate the ideal ofaVeronesevarietyPV*cP(Sym*V*)tothedecomposition givenin Exercise 15.33 ofSym*(Sym*¥). Forwhich kdothequadratie polyiomials vanishing theVeronese give anirreducible representation? Exercise 15.46. (For algebraic geometers and/or commutative algebraists) Justas thegroup PGL,C actsonthering$ofpolynomials onprojectivespace P*,preserving theideal oftheVeronese variety, soitacts onthat space of relations ontheideal (that is,inasmuch astheideal isgenerated byquadrics,thekernelofthemultiplication map1,(2)@S—5),andlikewiseontheentireminimal resolution ofthe ideal ofX.Show that this resolution has the form 4155. Representations ofGLC 2 ++R@S+R,@S—+1:QV@S, where altheR,arefinite-dimensional representations ofPGL,C, andidentify therepresentations R,inthespecific cases of (itherational normal curve inP?, (i)therational normal curve inP*,and (ii) theVeronese surface inPS. §15.5. Representations ofGL,C Wehave said that there islitte difference between representations ofGLC.andthoseofthesubgroup SL,.C ofmatrices ofdeterminant 1.Our object here istorecord thedifference, which, naturally enough, comes from thedeter- rminant: ifV=C*isthestandard representation, A'V istrivia forSL,C but notforGLC. Similarly, Vand /*'V* areisomorphic forSL,C butnot forGLC. Torelate representations ofSL,C andGL,C, wefirstneed todefine somerepresentations ofGL,C.Tobeginwith,letD,denotetheone-dimensionalrepresentation ofGL,C given bythekthpower ofthedeterminant. When kisnon-negative, D,=(A'V)®;Dsisthedual(D,)*ofD,.Next,notethattheirreducible representations ofSLC may belifted torepresentations ofGLC. intwoways. First, foranyindex a=(dy,..,d,)oflengthwemaytake®, tobethesubrepresentation ofthetensorproduct Sym"V @-- @Sym"H(A1V) @Sym™(NV) spanned bythehighest weight vector with weight a,L, +a,(L, +Ly)+ ay(Ly +°°"+Ly-a)—that is,thevector B=CE(eyAesencleyAAG ‘ThisrestrictstoSL4Ctogivetherepresentation Fy,wherea’=(24,..-y45-3); taking different vaiues ofa,amounts totensoring therepresentation with different factors Sym**(\'V) =(A°V)®* =D,,.Inparticular, wehave Death=Payot@Das which allows ustoextend thedefinition of©,toindices «with a,<0: we simply set Gooate=Pecutnts@Doe forlarge k. Alternatively,wemayconsidertheSchurfunctorS,appliedtothestandard representation VofGL,C,where Dm (aHH OyByHHyyyOya+My) Wewilldenote thisrepresentation 5,VofGL,Cby‘Wy;notethat Ways catgth=PaysovdyDe 2 15,a,Cand aC whichlikewiseallowsustodefine,foranyindex2with2,>2,2">Ayeven ifsome ofthe2,arenegative: wesimply take Pancake Wayttendgtt@Daa foranysufficiently large k. Asisnothard tosee,thetworepresentations ©,and‘Y,areisomorphic as representations ofGL,C:by§15.3 thelr restrictions toSL,C agree, soitsufices tocheck their restrictions tothecenter €*<GL,C, where each acts by multiplication by22=2!)Itisevenclearerthattherearenocoincidencesamong the®,(ie, ®,will beisomorphic toyifand only ifa=’: if ,=,,wemust have a,=ajfor{=I,..., —1,30 thestatement followsfromthenontrivialty ofD,fork#0.Thus,tocompleteourdescription ofthe irreducible inite-dimensional representations ofGLC,wejusthavetocheckthatwehave found them all.Wemay then express thecompleted result as Proposition 15.47. Every irreducible complex representation ofGLC isiso- ‘morphicto,forauniqueindex2=2yy2.52,with2y2Ay2°>hy(equi ‘alently, 10©,foraunique index a=as,..., 0,with dy, -.-.dy-y 20). Proor. Westart bygoing back tothecorresponding Liealgebras. The scalar matrices form aone-dimensional ideal €inglgC, andinfactql,C isaproduct ofLiealgebras: B= 6,0 xe. (15.48) {Inparticular, Cistheradicalofgl,C,andst,isthesemisimplepart.Itfollows {romProposition 9.17thateveryirreducible representation ofgl,Cisatensor productofanirreducible representation ofs!,Candaone-dimensional repre-sentation, Moreprecisely,letW=S,(C*)betherepresentation ofsl, deter- mined bythepartition 4(extended tosI,€ x€bymaking thesecond factor acttrivially). For weC,letL{w) betheone-dimensional representation of sl,€ xCwhich iszero onthefirst factor and multiplication bywonthe second; theproof ofProposition 9.17 shows that any irreducible representa- tion ofsl,C x€isisomorphic toatensor product W7,®L(w). The same is therefore true forthesimply connected’ group SL,C x€with thisLie algebra Wewrite GL,€ asaquotientmoduloadiscretesubgroupofthecenterof SLC xC: 1-+Ker(p)~»SLCx€4GLC+1, (15.49)wherep(gx2)=e*-g,s0thekernelofpisgeneratedbye*-1x(3),where5=2ai/n ‘Our task issimply toseewhich oftherepresentations W,@L(w) of SLC xCaretrivial onthekernel ofp.Now eI acts onS,C* bymulti "ForaproofthatSLCisimplyconnected,ne$23. $155. Representations ofGLC 233 plication bye%,whered =J.Ajrindeed, thisistrueontheentirerepresentation (€*) which contains $,C*. And —sactsonL{w) bymultiplication bye”*,s0 ex (=s)acts onthetensor product bymultiplication bye"™. Thetensor product is,therefore, trivial ontheKernel ofpprecisely when sd—sw€2niZ, ie, when w=DAtkn forsome integer k. Weclaim finally that any representation W;@Lw) satisfying thiscondi- tionisthe pullback viapofarepresentation ¥onGL,C. Infact, itisnothard to.ee thatitisthepullback oftherepresentation ,, 4.2+4:thetwoclearly restrict tothesame representation onSLC, andtheir restrictions to€are justmultiplication bye**=etn», fs} Exercise 15.50. Show thatthedual oftherepresentation ,which isiso- morphic to5,(V*)is therepresentation M4.) Exercise 15.51*, Show thatifp:GLC -»GL(W)is arepresentation (assumed tobeholomorphic), then Wdecomposes into adirect sum ofirreducible representations. Exercise 15.52*. Show thaltheHermite reciprocity isomorphism ofExercise 11.34 isanisomorphism over GLC, notjustover SL3€. More Remarks onWeyl’s Construction ‘Weclose outthislecture bylooking once more attheWeyl construction of these representations ofGL(V). This willincludearealization“bygenerators. and relations,” aswell 2sgivinganaturalbasisforeachrepresentation. First, itmay beilluminating—and itwill beuseful later—to look more closely at how 5, sitsinV®. Wewant torealize S,V asasubspace ofthesubspace Syme (AV) @Sym™(AV) @--@Syme(V) cH, ‘whereaisthenumberofcolumnsoftheYoungdiagramofoflengthi(and kisthenumberofrows).Thisspaceisembedded inV®inthenaturalway:from lefttoright, afactor Sym*(/PV) isembedded inthecorresponding V°"* bymappingasymmetric productofexteriorproducts (CORE Se Ce (ieA204°"AOa) 0 L8B(9)(O19,001)BBUgg19)@°°B(eer) BBPeso.vesthesumoverpeS,andg=(q1,-...4a)€ Syx***xGy,Inotherwords,one 234 15.aCandsi firstsymmetrizes bypermuting columnsofthesamelength,andthenperformsanalternating symmetrizer oneach column. Lettinga=(a,,...,a4),letA*(V)denotethistensorproductofsymmetric powersofexterior powers, Le,set AMV=Sym™(NV)® Symm™(N-*V)@--- @Sym™(V). ‘WewanttorealizeSVasasubspaceofA*¥,Todothisweusetheconstruc- tionofSVasV&*c,,wherec,isaYoungsymmetrizer; togetcompatibility with theembedding ofA*V wehave just made, weusethetableau which ‘numbers thecolumns from toptobottom, then lefttoright. vfefele ayaa s[7peet aes ae? uel pany my ee saad Wetake j=2!=(yt; 2-2 1,>O}to betheconjugateof4Thesymmetrizer cxisaproduct a,"by, where a,=S.e,, thesumover allpinthesubgroup P=G,,x-+" xS,,ofG,preserving therows,b,=Y.sgn(q)q,thesumover thesubgroup Q=&,,x-~- xS,,preserving thecolumns, asdescribed in Lecture 4.Thesymmetrizing byrows canbedone intwosteps asfollows. ‘There isasubgroup R=G xx, ofP,which consists ofpermutations that move allentries ofeach column tothesame position insome column ofthesame length; inother words,permutations inRaredetermined bypermuting columnswhichhavethesamelength. (Intheillustration, R=(1,(46)(57)},) Set a=Zeince, NowifwedefineatobeYep,wherethesumisoveranysetofrepresentatives inPforthefeftcosetsP/R,thentherowsymmetrizer a,istheproductofanda3.So SAV)=(VR G3)-a4-by. ‘The point isthat, bywhat wehave justseen, VS a,b =AV. 153. Representations ofGLC 235 Since V4, isasubspace ofV®, itsimage S,(V) byab, isasubspace of ‘A*(V), a8weclaimed, ‘There isasimple way toconstruct alltherepresentations $,V ofGL{V) atonce. Infact, thedirect sum ofalltherepresentations $Y, over all(non- negative) partitions 2,eanbemade into acommutative, graded ring, whichwedenotebyS'orS'(V),withsimplegenerators andrelations.Thisissimitartothefactthatthesymmetric algebra Sym'V =@)Sym*V andtheexterior algebra NV=)/*Vareeasiertodescribethantheindividualgradedpieces, and ithassome ofthesimilar advantages forstudying alltherepresen- tations atonce. This algebra hasappeared and reappeared frequently, cf. [H-PJ; theconstruction wegive isessentially that ofTowber [Tow1}. Toconstruct$'(V),startwiththesymmetric algebraonthesumofall the positive exterior products ofV:set AW) =SymVONVORVO--® NV) =@_Sym™(AV)@-- @Sym4NV)@ Sym"(V), thesum over allntuples a,,....dyofnon-negative integers.SoA°(V)isthe direct sum ofthe4*(V) justconsidered. Thering S'=S'(V) isdefined tobe thequotient ofthis ring4(V) modulo thegraded, two-sided ideal generated byallelements (“Pliicker relations") oftheform (oy, AAa AAW) -foee ee Oe) (1533) forallp>q>1andall055-4UpsWhy«oo.WyeV.(Ip=g,thisisanele-mentofSym?(A°V}; ifp>g,itisinWV@AV=Sym!(AP¥)@Sym’(A‘V).Notethatthemultiplication in$(V)comes entirelyfromitsbeingasymmetric Algebra and does notinvolve thewedge products inNV.) Exercise 15.54*. Show that "contains allelements ofthe form (opA Oo A Amy) HLOp A AWA AM AAG) (80, AAA Mey AEA Ma) forallp>q21>Uandall5.205Up,WhyoorMywherethesumisoverall<i <iy<"~* <i,<p,and theelements w,,..., w,areinserted atthe corresponding places invyA°"*6tp Remark. You canavoid thisexercise bysimply taking theelements inthe ‘exercise asdefining generators fortheideal J".When p=q=r,thecalcula- 236 18,slyCanda, tion ofExercise 15.54 shows that therelation (0,a"" 61,)-(m) A" 4mp) =(my8°A.W,)-(0) A= A)follows fromthegenerating equations forI’. Inparticular, thiscommutativity shows thatonecould define S'(V) tobethe fulltensor algebra onV®\?V. ®---@AV modulo theideal generated by thesame generators. ‘Thealgebra S'{V) isthedirect sumoftheimages 5*(V) ofthesummands AV),Lete1,..-,€, beabasisforV.Wewillconstruct abasisforS*(V),with‘abasis clement e,forevery semistandard tableau Tonthepartition 2which corresponds toa.Recall thatasemistandard tableau isanumbering oftheboxesoftheYoungdiagramwiththeintegers{,...,,insuchawaythattheentries ineach row arenondecreasing, and theentries ineach column are strictly increasing. LetT(i,j) betheentry ofTintheithrow and thejth column. Define eytobetheimage in$*(V) oftheelement eran eran6°erp€SMV @Sym"), ive,wedge together thebasis elements corresponding totheentries inthe columns, andmultiply theresults in$(V). Proposition 15.58. (1)Theprojection from A*(V) toS*(V) maps thesubspace S,(V) isomorphically onto S*(V).. (2)TheeyforTasemistandard tableau onAform abasis for$*(V). Proor. Weshow first that theelements e,span S*(V). Itisclear that thee span ifweallow alltableaux Tthat number theboxes of2with integers between |andnwithstrictlyincreasing columns, forsuchelements spanbeforedividingbytheidealI".Weordersuchtableauxbylistingtheirentriescolumn bycolumn, from lefttoright and top tobottom, and using the reverse lexicographic order: T’>Tifthelastentrywheretheydifferhasalargerentry forT’thanfor7:IfTisnotsemistandard, therewillbetwosuccessivecolumns ofT,saythejthand(j+1)st,inwhichwehaveT(r,j)>T(r,+1)forsomerr.Itsuffices toshow how touserelations ins towrite ¢,asalinearcombination ofelements¢,.withT”>T.ForthisweusetherelationinExercise 15.54, with 0=ery. for 1<i< p=ppand w= ery) forUSiSq=sss,tointerchange thefirstrofthe{w,}withsubsetsofrofthe {nj}.Thetermsontheright-hand sideofthe relation will allcorrespond to tableaux 7”inwhichtherfirstentriesin the(j+1)stcolumnofTarereplacedbyroftheentiesinthejthcolumn,andarenototherwisechangedbeyondthethcolumn.AllofthesearelargerthanTintheordering, whichprovesthe assertion Itispossible togive adirect proof that theeycorresponding tosemi-standard tableauxTarelinearlyindependent (seeTow1]), butwecangetby with less. Among thesemistandard tableaux on2there isasmallest one Ty ‘whose ithrowisfiled withtheinteger i.Weneed toknow thatey,inotze 8155. Representations ofGLC 27 inS%.This iseasy toseedirectly. Infact, therelations among thee,in FAY) arespanned bythose obtained bysubstituting relements from ‘some column ofsome Ttoanearlier column, asinthepreceding paragraph, Such willnever involve thegenerator ¢,,unless the7’that isused isTq,andinthiscase,theresultingelementofiszero.Sinceey,occursinnonontrivialrelation, itsimage inS*cannot vanish. Since e,,comes from 5,(V), itfollows thattheprojection from 5,(V) to S*(V) isnotzero. Since this projection isamapping ofrepresentations ofSL(V)itfollowsthatS*(V)mustcontainacopyofthe irreducible representa~ tion §,(V). Weknow from Theorem 6.3and Exercise A.3) that thedimension‘ofS,(V)isthenumberofsemistandard tableauxon2.Sincewehaveprovedthat thedimension of§*(V) isatmost this number, theprojection from 5,(V)toS*(P)mustbesurjective, andsinceS,(V)isirreducible, itmustbeinjectiveaswell, andthee,forTasemistandard tableay on2must form abasis, asasserted. o Note thatthisproposition gives another description oftherepresentations S,(V), asthequotient ofthespace AV) bythesubspace generated bythe“Picker”relations(15.53). Exercise 15.56. Show that, ifthe factor /*¥isomittedfromtheconstruction, theresulting algebra isthedirect sum ofallirreducible representations of SL(V) =SLC. Itisremarkable thatalltherepresentations $,(C") ofGL,C were written down byDeruyts (following Clebsch) acentury ago, before representation theory was born, asinthefollowing exercise. Exercise 15.57*. LetX=(x,,) beannxnmatrix ofindeterminants, The groupG=GL,CactsonthepolynomialringC[x,,]byg-x1,)=Dhar0% forg=(«,,)€GL,C.ForanytableauTontheYoungdiagramofconsisting, oftheintegers from 1ton,strictly increasing inthecolumns, leteybethe product ofminors constructed from X,oneforeach column, asfollows: ifthecolumnofThaslengthjy,formtheminorusingthefirstcolumns,andusetherows that arenumbered bytheentries ofthecolumn ofT:LetD,bethe subspace ofC[x,,] spanned bythese e,,where disthenumber partitioned byAShow that: ()D, ispreserved byGLC; (i)theep,where Tissemi- standard, form abasis forD,:(li)D,isisomorphic toS,(C") LECTURE 16 Symplectic LieAlgebras Inthislecture wedoforthesymplectic Liealgebras exactly what wedidforthespecial linearonesjn§15.1andmostof§15.2:wewillfirstdescribe ingeneralthestructure ofasymplectic Liealgebra(thatis,giveaCartansubalgebra, findtheroots,describetheKilling form, andsoon).Wewillthenwork outinsome detail therepresentations ofthespeciealgbrasp.C.Asinthecaeofthecorresponding analysisofthespeciallinear Liealgebras, thins completely elementary. $164:ThestructureofSp,,€andsp34C$162 Representations ofspuC §16.1. The Structure ofSp,,€ and sp;,C Let¥bea2n-dimensional complex vector space, and OVxVAC, 1nondegenerate, skew-symmetric bilinearformonV.Thesymplectic Liegroup Sp;,C isthen defined tobethegroup ofautomorphisms Aof¥preservingQ—thatis,suchthatQ(4v,Aw)=Q(o,w)forallv,we¥—andthesymplectic Liealgebra spz,C correspondingly consists ofendomorphisms A:VV satisfying (40, #)+Ole, Aw) =0 forallvandw€¥.Clearly,theisomorphism classesoftheabstractgroupandLiealgebra donotdepend onthepasticular choice ofQ;butinorder tobe able (owrite down elements ofboth explicitly wewill, fortheremainder ofourdiscussion, take@tobethebilinearformgiven,intermsofabasise,,.... G64,TheStructofSpaandsy. 29 €2_forV,by Ole etnd= 1, ein ei)=—1, and Qe,¢)=0 ifjeitm ‘The bilinear form Qmay beexpreted as Q(x, y)="xMey, where Misthe2nx2nmatrix given inblock form as 1),wo(L, op thegroupSpz,Cisthustheproupof2nx2nmatricesAsaitying M='A-M-A andtheLiealgebra sp2,€ correspondingly thespace ofmatrices Xsatisfying the relation XM+M°X=0. (16.1) Writing a2nx2nmatrix Xinblock form a8 ABx-(¢9) we have . ='c Wxwe(“ 2) and cp wx-(S, sothat thisrelation isequivalent tosaying that theoff-diagonal blocks Band CofXaresymmetric, andthediagonal blocks Aand DofXarenegative transposes ofeach other. With this said, there iscertainly anobvious candidate forCartan sub- algebra binsp3,€, namely thesubalgebra ofmatrices diagonal inthis representation; infact, thisworks,asweshallseeshortly.Thesubalgebrahis, thusspanned bythen2nx2n matrices Hf,=E,,~Eysgeys whose action on Vistofixe;,send e,4,toitsnegative, andkillalltheremaining basis vectors; wewillcorrespondingly take asbasis forthedual vector space h*thedual basis Ly,where (Ly,Hi)~4,. ‘Wehave already seen how thediagonal matrices actonthealgebra ofall ‘matrices, sothat itiseasy todescribe theaction ofhong.For example, for 40 16.Symplectc LieAlgebras 1.i,j<nthematrixE,,€plasiscarriedintoitselfundertheadjointactionofH,,into minus itself bytheaction ofH,,andto0byalltheother Hy;andthesameistrueofthematrixE,.,,.;Theelement Xi Big~Enssass ©axl isthusaneigenvector fortheactionofb,witheigenvalue L,—Ly.Similarly,fori#jweseethatthematricesB44,andF,q4,arecarriedintothemselvesbyHandH,andkilledbyalltheotherHa;andlikewiseE,,,,andE4,,arecach carried into their negatives byH,andH,andkilled bytheothers. Thus, the elements Yay Bunty +Enns and Zs=Basis+Basseareeigenvectors fortheactionofb,witheigenvalues Ly+L,and—L,~Ly,respectively. Finally, when f=jthesame calculation shows that Ey,us iS doubled byH,andkilled byallother H,;andlikewise E,,,, issenttominus twice itself by11,andto0bytheothers. Thus, theelements U,= Exess and Vim Bevis areeigenvectors with eigenvalues 2L,and—2L,, respectively. Insum, then, theroots oftheLiealgebra sp,€ arethevectors +L,+Lye5*.Inthefirstcasen=1,ofcoursewejustgettherootdiagramofsls,which isthesame algebra asspC.Incasen=2,wehave thediagram XE . ‘Asinthecase ofthespecial linear Liealgebras, probably theeasiest way todetermine theKilling form onsp5,€ (atleast uptoscalars) istouseits $16.1.TheStructure ofSpy_Cand*p;,C Pr) invariance undertheautomorphisms ofsp,,Cpreserving b.Forexample, whave theautomorphisms ofsp3,C induced bypermutations ofthebasi vectors e,of¥:forany permutation oof{1,2,...,1} wecan define a: automorphism ofVpreserving @bysending ,{0ennAdeye10Eysn 3M thisinduces anautomorphism ofsp24C preserving handcarrying H,toHq, Also, foranyiwecandefine aninvolution of¥—and thereby ofsp,,C—b: sending €,t0ey44, paitO—€,, andalltheother basis vectors tothemselves thiswillhave theeffect ofsending H,to—H,andpreserving alltheother H, Now, theKilling form on}must beinvariant under these automorphisms fromthefirstbatchitfollowsthatforsomepairofconstants «and)wemushave Billy Hy)=a and BUH, H,)=f fori #j; fromthesecondbatchitfollowsthat,infact,f=0.Thus,Bisjustamultipleofthestandard quadratic formB(H,,H,)=6,andthedualformcorrespond:ingly amultipleofB(L,,L,)=6:0thattheanglesinthediagramaboveare correct. Alsoasinthecaseofsl,C,onecanalsocompute theKillingformdirectlyfrom thedefinition: B(H, H’)=¥a(H)a(H"), thesumover allroots @.For H=Y.qH,and H'=b,, thisgives B(H, H’)asasum 4, 4)+2,2a)(2bi) (a,—aby— Flotath+)+28aneann+Fcaach~) which simplifies to BU, H')=(4n+USby (163) Our next jobistolocate thedistinguished copies s,ofslxC, and the corresponding elements 11,€.This iscompletely straightforward. Westart with theeigenvalues L,—LyandL,—L,corresponding totheelements X,,, andX;,.;wehave Da Mud =CEs ~Bsssnet Bua~Ensines] =CEays Ey) +[EpsunsisExsinss] =Bir Ey+Bassnty—Bastaot =Hy ‘Thus,thedistinguished element H;,.., isamultiple ofH,—H,.Toseewhat multiple, recall thatH,,-1, should act’on X;,,bymultiplication by2andon X,,bymultiplication by~-2;since wehave d(H, —HY) =(Lr—LM —HY)Xay =Xp mw 16,SymplectcLieAlgebras wweconclude that Hau, =He Hy Next consider thepairofopposite eigenvalues L, +L,and ~Ly—Ly corresponding totheeigenvectors ¥,andZ,,.Wehave Chas Zi) =Benes +Exner Envy +Bosse] =Lene Bassa +[joo Eras] =Bu Bnsyats+EssEnsuast =H+ty We cateulate then d(H,+HY)=(Ly+LE+HD)Yay a2Ky s0we have Aya, =H+ Hy andsimilarly Haren =Th Hy Finally, welook atthepair ofeigenvalues +21, coming from theeigen-vectorsU;andF,.TocompletethespanofU,andV;toacopyofsl,Cweadd [UV =cesi Easd =Eu Ensiast =H Since ad(Hi)(U) =LH) U, =2U, wweconclude thatthedistinguished element H,, isHy,andlikewise Hs, = —Hy, Thus, thedistinguished elements {H,} ©)are(4H, +Hy+H); in particular, theweight lattice Ayoflinear forms onjyintegral omalltheHis‘exactlythelatticeofintegrallinearcombinations oftheLy.InDiagram(162)forexamplethsisjustthelatticeofintersections ofthehorizontalandvertical linesdrawn;observethatforalltheindex[AyAx]oftherootlatticeintheweight lattice isjust 2.[Nextweconsiderthegroupofsymmetries oftheweightsofanarbitraryrepresentation ofsp,C. For each root «weletW,betheinvolution in 1fixing thehyperplane 9,given by(H,, L)=0and acting as—Ionthe line spanned bya;weobserve inthis case that, asweclaimed will betrue in ‘general, thelinegenerated byaisperpendicular tothehyperplane 0,sothat theinvolution isjust areflection inthis plane. Intheease m=2,forexample, 816.1,TheStrwctureofSp,€andaps, Py wegetthedihedral groupgenerated byreflections aroundthefourlinesdrawnthrough theorigin: SANZahihyBeer,PSSIEPPR LLP sothat theweight diagram ofarepresentation ofsp,C will look like an ‘octagon ingeneral, or(insome cases) asquare. Ingeneral, reflection intheplane 5), given by<H,,L)=0willsimply reverse thesignofL,while leaving theother L,fixed; reflection intheplane <H,— Hy,L)=Owillexchange LyandL,andleave theremaining Lyalone ‘The Weyl group 9Bactsasthefullautomorphism group ofthelines spanned bytheL,andfitsinto asequence 1(Zpzy +B46,41 Note that thesequence splits: %isasemidirect product ofS,and (Z/2Z)".(Thisisaspecialcaseofawreathproduct,Inparticular theorderofWis2n!. ‘Wecanchoose apositive direction asbefore: ULGLa)=Cydy+0+yyy Ce>CQ>>>O ‘Thepositive roots arethen RY=(Lt bhi (lihep (164) withprimitive positive roots {L;—Lys,}y=1,..,.-1 2nd2L,.Thecorresponding (closed) Weyl chamber is W=(a,Ly +a;Ly+0"+aby,2a,2-24, 20};(16.5) note that the walls ofthis chamber—the cones {Zabyay>>a,=yyy>0>a,>OF and (Cabsa,>a,>">a,=0} linthehyperplanes, nd9,,, perpendicular totheprimitive positive ‘ornegative roots, asexpecied. 24 16.Symplecti LieAlgebras §16.2. Representations ofsp. Letusconsider nowtherepresentationsofthealgebrasp,specifically.Recall that, with the choiceofWeylchamber asabove,thereisauniqueirreducible representationI,ofsp,Cwithhighestweight«foranyaintheintersection oftheclosed Weyl chamber #with theweight lattice: thatis,foreach lattice vector intheshaded region inthediagram D<Dx<P< PepeVANE AN ee Xxx x} RRRBereaX>x>x<p hoy ch ges ght ver canbemite 3son-sepie inte linear combination ofL,and L,+L; forsimplicity wewilljust writeTsfortheirreducible representation Typ,su,+z»Withhighestweight endomorshisms ofthefour-dimensional vectorspaceVsthefouranand basis vectors ¢,,¢2,¢3, ande,areeigenvectors with eigenvalues L,,L2,—Ly, and —Ls,respectively, sothattheweightdiagramofVis, Sh Dp] XxX VAG 4162. Representations ofsp us Visjusttherepresentation Tointhenotation above.Notethatthedualofthis representation isisomorphic toit,which wecan seeeither from thesymmetryoftheweightdiagram,ordirectlyfromthefactthatthecorrespond- ing group representation preserves abilinear form VxV-+C giving an identiication of ¥with V* ‘Thenextrepresentation toconsider istheexterior square A?V.Theweights ofA?¥,thepairwise sumsofdistinct weights ofV,arejustthelinearforms +L;+L,(each appearing once) and0(appearing twice, asLy—Lyand L,—L,),80 thatitsweight diagram looks like KKKDX Tx[xKOKbelxhsVSSSKo<IxhPVAAN ‘Clearly thisrepresentation isnotirreducible. Wecanseethisfrom theweight diagram, using Observation 14.16: there isonly oneway ofgetting tothe ‘weight space 0from thehighest weight L,+L3bysuccessive applications of theprimitive negative rootspacesg-1,+1,(spanned byX,,,=E,.;~Es,«)andg_3., (spanned byV;=E,,,)—that is,byapplying firstV;,which takes ‘youtotheweight space ofL,—L,andthen X;,,—and sothedimension ofthezeroweightspaceintheirreduciblerepresentation I,withhighestweight L,+L,must beone. Ofcourse, weknow inanyevent that \?V cannot be itreducible: thecorresponding group action ofSp,C onVbydefinition preserves theskew form Q¢\?V* =A?V. Either way, weconclude thatwe have adirect sum decomposition NV=wee, where Wistheitreducible, fve-dimensional representation ofep<C with highestweightL,+L,—inournotation, To,,—-andweightdiagram . mt xhEPReRPLRY Letusconsider nextsomedegree2tensorsinVandW.Tobeginwith,we‘canwritedowntheweightdiagram fortherepresentation Sym?V; theweightsbeingjustthepairwise sumsoftheweights ofV,thediagram is BhsRepKaanLLL ‘Thislooksliketheweightdiagramoftheasointrepression seaindeedtam‘ention161eningthesymplecticieletsthatte sp.cHom(h, V)=V@V" =VV isjustthesubspace Sym?V <[email protected], Sym*V istheirreducible representation T,owith highest weight 2L. ‘Next, consider thesymmetric square Sym?W, which hasweight diagram $162. Representations ofspeC 247 ehh.BERRREY WEAVERANAiSNE ranePRREERE Toseeifthisisirreduciblewefirstlookattheweightdiagram:thistimethere arethreewaysofgettingfromtheweightspacewithhighestweight2L,+2L, tothespace ofweight 0bysuccessively applying X;,, =Es, —Es,« andVY;=E,g,2,S0ifwewanttoproceedbythismethodweareforcedtodoalittlecalculation, which weleave asExercise 16.7. Alternatively, wecanseedirectlythatSym?Wdecomposes: thenaturalmap. given bywedge product NV@NV+AMV =o ‘issymmetric, and sofactors togive amap Sym4{A¥V)) +. Moreover, since thismap iswelldefined uptoscalars —in particular, itdoes notdepend onthechoice ofskew form Q—it cannot contain thesubspace ‘Sym?W cSym?(A?¥)) initskernel, sothatitrestricts togiveasurjection g:Sym? W—C. ‘This approach would appear toleave two possibilities open: either the kernel ofthis map isirreducible, oritisthedirect sum ofanirreducible representation and afurther trivial summand, Infact, however, from the principle that anirreducible representation cannot have two independent invariant bilinear forms, weseethat Sym?W cancontain atmost onetrivial summand, and sotheformer alternative must hold, i.c., wehave Sym'W=Ty,2OC. (16.6) Exercise 16.7%. Prove (166) directly, byshowing that ifvis@highest weight vector, then thethree vectors X;.,VX2,1¥i0, Xz.1X2,1¥a¥av, andWX,.1X;,,¥e0 spanatwo-dimensional subspaceofthekernelofo. Exercise 16.8, Verify that A?W =Sym*V. Thesignificance ofthisisomor- phism willbedeveloped further inLecture 18. Lastly, consider thetensor product [email protected], itsweight diagram: <b> x<PuxXp ‘This obviously must contain theirreducible representation I, with highest weight 2L,+L,;butitcannot beirreducible, foreitheroftworeasons. First, +L, with multiplicity atmost 2,sothat V@Wmust contain atleast onecopy.ofthe representation V.Alternatively, wehave anatural map given bywedge cies AVONV AV =Vt=V; andsince thismap does notdepend onthechoice ofskew form Q,itmust restrict togive anonzero (and hence surjective) map oVOW. Exercise 16.9. Show that thekernel ofthismap isirreducible, andhence that we have: V@Ww=",,0V. What about more general tensors? Tobegin with, note that wehave established theexistence halfofthestandardexistenceanduniqueness theorem (14.18) inthecaseofsp,C: theirreducible representation I,may befound somewhere inthetensorproduct Sym*V@Sym*W. Thequestion thatremains: products decompose. This will be,asitwas inthe case ofsi,C, nearly tantamount (modulo thecombinatorics needed tocount themultiplicity with which thetensor product Sym*V@Sym*Wassumes eachofitseigenvalues) Letusstart with thesimplest case, namely, therepresentations Sym*¥.‘Thesehaveweightdiagramasequenceofnesteddiamonds D,withverticesat $162. Representations ofspa 249 Moreover, iisnothard tocalculate themultiplicities ofSym*V: themulti plicity ontheouter diamond D,isone, ofcourse; andthen themultiplicities will increase byone onsuccessive rings, sothat themultiplicity along the diamond D,will bet Exercise 16.10. Using thetechniques ofLecture 13,show that therepresenta-tionsSym*Vareirreducible. ‘The next simplest representations, naturally enough, arethesyinmetric powers Sym'W ofW.These have eigenvalue diagrams intheshape ofa sequence ofsquares S,with vertices atb(Ly +L3),(b— 1)(Ly +La),andso e+a—— foe|| o—e--— Here, however, themultipicties increase inarather strange way: they grow quadratically, butonlyonevery other ring. Explicitly, themultiplicity willbe foneontheouter tworings, then 3onthenext tworings, 6onthenext two; ingeneral, itwillbe(i+1)/2onthe(21—I)stand(2i)th squares Sy). and Sw 20 16.Spee LiAgee Exercise 16.11. Show that contraction with theskew form @¢Sym?W* introduced inthediscussion ofSym?W above determines asurjection from Sym*W onto Sym*-2W, andthat thekernel ofthismap istheirreducible Srrcactnton Fay with Mghe welght Bye Lah Show tat themal Pieliceal syaston theestes Sev and dco above Wewillnishbyanalyzing, naivelyandindetail,oneexampleofarepresen-tation T.,,with aandbboth nonzero, namely, F,,;one thingwemayobserve(onthebasisofthisexampleisthatthereisnotasimilarlysimplepatternto themultiplicities oftherepresentations F,,with general aandb.Tocarry out‘ouranalysis,westartofcourse withtheproduct [email protected] draw theweight diagram forthis representation; drawing only one-cighth of theplane andindicating multiplicities bynumbers, itis 2S <xKOSVGNSAB ‘Weknow thattherepresentation Sym?V @Wcontains copy oftheirreducible representation T3,, with highest weight 2L,+(L, +Ly): and wecansee imoedintly orth diag hat ronol goal resample acon take theweight 2L,with multiplicity atmost 2(ifv¢Tis itshighest weight vector, thecorresponding weightspace(Ts,,)az,©Fa,Willbespannedbythe two vectors Xp,,(V;(0)) and V5(Xp,(0))} since itcannot contain acopy of therepresentation I,(the multiplicity oftheweight 2(L, +L,)being just one) itfollows that Sym?V@Wmustcontainacopyoftherepresentation Tyo =Sym*V, ‘Wecan, inthisway, narrow down thelistofpossibilities agood deal. For example, ,cannothavemultiplicity justoneateachoftheweights2L,and Ly+Ly:ifitdid,Sym?@WwouldhavetocontaintwocopiesofSym?V ‘and afurther twocopies ofWtomake upthemultiplicity atLy+Ly;but sinceOmustappearasaweightof[>,,thiswouldgiveatotalmultiplicityof atleast 7fortheweight 0inSym?¥@W.Similarly,F,,cannothavemulti- plicity 1at2L,and2atL,+Ly:wewould then have twocopies ofSym?V andoneofWinSym?V@W;andsincethemultiplicityofOinT,,willin thr cae beatnt? (Ocng rei than egal Lathe mlicy ot 1 ty ison ase aye mui oft eas orthewigan0 $162.Representations ofspa 251 inSym*V @W.Itfollows thatSym?V @Wmust contain exactly onecopy of ‘Sym?V; andsince themultiplicity ofLy+L,inT,,isatmost 3,itfollows thatSym?¥ @Wwillcontain atleast onecopy ofTo,=Was wel. Exercise 16.12. Prove, independently oftheabove analysis, that Sym?V@W ‘must contain @copy ofSym?V andacopy ofWbylooking atthemap o:Sym’VQW+V@V obtained bysending U-0@lwA2U@Gorwr2)+o@GWAWA2 where weateidentifying AVwith thedual space V*anddenoting byG:V*-Vtheisomorphism inducedbytheskewformQonV.Specifically,show that theimage ofthis map iscomplementary tothelinespanned bythe clementQeMV*=VcV@Y. ‘Theabove leaves uswith exactly twopossibilities fortheweights ofF,, weknow that themultiplicity of2L,inTis exactly 2;s0either themultiplicities ofLy+L,andOinT,,areboth3andwehave Sym?V@W=13,1@Sym'V @Ws orthemultiplicities ofLy+Lyand0inI,areboth 2andwehave Sym?¥ @W=T3,,@Sym'V. OW Exercise16.13.Showthattheformerofthesetwopossibilities actuallyoccurs,by (2)Showing thatifvisthehighest weight vector inT3,, ¢Sym?V.@ W, thentheimages (X,:)*Va(0) X3,1VaX2,s(0h andV(X, )?0areindependent, and (redundantly) (b)Showing thattherepresentation Sym? @Wcontains onlyonehighestweightvectorofweightLy+La ‘Theweight diagram ofI,istherefore mayciate ERRPERER 232 16.Symplectc LieAlgebras We seefrom allthis that, inparticular, theweights ofthe irreduciblerepresentations ofsp,Carenotconstantontheringsoftheirweightdiagrams, Exercise 16.14. Analyze therepresentation V@Sym?W ofspxC. Find in particular themultiplicities oftherepresentation I,.. Exercise 16.15. Analyze therepresentation Sym?V@Sym?Wofsp4C.Find inparticular themultiplicities oftherepresentation T's,2. LECTURE 17 sp6C and sp,,C Inthefiattwosectionsofthislecturewecompleteourclasifcation oftherepresenta:tionsofthesymplecticLiealgebras:wedescribeindetailtheexampleofsp,theasketch therepresentation theory ofsymplectic Liealgebras ingeneral, inparticularprovingtheexistencepartofTheorem14.18forsp,athenalsectionwedescribe fnanalogforthesymplecticalgebrasoftheconsruetion givenin153oftheitredue-iblerepresentations ofthe special near algebras viaWey’ construction, though we Postpone giving analogous formulas forthedecomposition oftensor products of irreducible representations. Sections 17.1and 17.2arecompletely elementary, given thebynow standard multilinear algebca ofAppendix B.Section 173, like 6153, requires familiarity with thecontents ofLecture 6andAppendix A;but,likethatsectionitcanbeskippedwithoutaffectingmostoftherestofthebook.$17.1:Representations ofsp4C4172: RepresentationsofthesymplecticLiealgebrasingeneral 417. WeyT' construction forsymplectic groups §17.1. Representations ofsp. ‘Aswehave seen, theCartan algebra hofspe isthree-dimensional, with the linear functionals Ly,Lz,and Lsforming anorthonormal basis interms oftheKillingform;andtherootsofspgCarethenthe18vectors+Ly+Ly.Wecandrawthisintermsofa“reference cube”inh*withfacescentered atthepoints+1,thevectors+L,+L,withi#jarethenthemidpoints ofedgesofthisreference cube and thevectors +2L, themidpoints ofthefaces ofa cube twice aslarge. Alternatively, wecandraw areference octahedron with vertices atthevectors +2l,; theroots +L,+L,with i#Jwillthen bethe 2s 17.apsCandown midpoints oftheedges ofthisoctahedron: or,ifweinclude thereference cube aswell, as Al<|ins anyae, ‘This lastdingram, however ineplly drawn, suggests acomparison with the rootdiagramofs1,C;infactthe12rootsofspgCoftheform+L,+L,for14jarecongruent tothe 12roots ofsC. Inparticular, theWeyl group ofn4CwilbegeneratedbytheWeylgroupofel,plusanyoftheadiionalthree reflections intheplanes perpendicular totheL,(ie., theplanes parallel tothefaces ofthereference cube intheroot diagram ofeither Liealgebra). Wecanindicate theplanes perpencicular totheroots ofsp4C bydrawing where they cross thevisible part ofthereference cube: S174,Repeetton of4p4¢ ass Weseefrom thisthat theeffect oftheadditional reflections intheWeyl groupofsp¢€ontheWeylchamberofs{,issimplytocutitinhalf;whereas. theWeyl chamber ofsI,€lookedlike '\ iLo theWeyl chamber ofsp.C willlook likejusttheupper halfofthisregion: a I Interms ofthereference octahedron, thisisthecone over onepart ofthe barycentric subdivision ofaface: 286 17epg and sp. Sea ofifwerotate90°aroundtheverticalaxisinanattempttomakethepictureclearer, m Weshould remark before proceeding that thecomparison between theroot systems ofthespecial linear algebra sl,C andthesymplectic algebra spCis peculiartothiscase;ingeneral,therootsystemsofst,,,€CandspyCwillear nosuch similarity. ‘Aswesaw inthepreceding lecture, theweight lattice ofspe consistssimplyoftheintegrallinearcombinations oftheweightsL,.Inparticular,theintersection ofthe weight lattice with theclosed Weyl chamber chosen above willconsist exactly ofintegral linear combinationsa,Ly+agL+a3Lywith 4,2032ay>0.ByOurgeneralexistenceanduniquenesstheorem,then, forevery triple (a,),c) ofnon-negative integers there will exist auniqueinreducible representation ofspgCwithhighestweightaL,+(Ly+La)+ $17.1, Representations ofpg 2 eLy +Ly+Ly) = (a+b +Ly +(b+ Ly +cL;wewilldenotethisre resentationbyI...andwilldemonstrate itsexistenceinthefollowing. ‘Westart byconsidering thestandard representation ofspgC onV=( ‘The eigenvectors oftheaction offon ¥arejust thestandard basis vecto ,,and these have eigenvalues +L,,s0 thattheweight diagram ofVlooksli! themidpoints ofthefaces ofthereference cube (orthevertices of: octahedron one-half thesizeofthe reference octahedron): ty Inparticular, Vistherepresentation T,0,0- Since wearegoing towant (ofind arepresentation with highest weigl Ly+Ly,thenatural thing tolook atnext isthesecond exterior power A?ofthestandardrepresentation. Thiswillhaveweightsthepairwisesum«distinct weights of¥,orinother words the12weights +L,+L,with i¥ andtheweight 0taken three times. This isnotirreducible: bydefinition thactionofspeConthestandardrepresentation preservesaskewform,s0the therepresentation on/?Vwillhave atrivial summand, Ontheother hanc theskew form onVpreserved byspgC, and hence that trivial summand « XY, isunique; andsince allthenonzero weights ofA'V occur with mult plicty 1andareconjugate under theWeyl group, itfollows that thecompletmentWofthetrivialrepresentation inA*Visirreducible.SoW=To,.o Asinprevious examples, wecanalsoseethat/?V isnotirreducible b using thefact(Observation 14.16) that theirreducible representation To,, ‘withhighestweightLy+L;willbegenerated byapplyingtoasinglehighe:weight vector vtherootspaces g,,-1,» 8i,-r4» ANd8-21, Corresponding t+ primitive negative roots. Wecanthen verify that intheirreducible representa tion Wwith highest weight L,+La,there areonly three ways ofgoing fron thehighest weight space tothezero weight space bysuccessive application 0 these roots spaces: wecango Ly+laky Rett LytLy+l;-Ly +0 28 17.ap4C andoyn Exercise 17.3. Verify this, and also verify that thelower two routes tothe zeto-weight space inA?V yield thesame nonzero vector, andthattheupperrouteyieldsanindependent elementofAV,sothat0doesindeedoccurwith multiplicity 2asaweight ofTo,s,0- Tocontinue, welook nextatthethird exterior power A’ofthestandard representation; weknow that this will contain acopy oftheirreducible representation To, withhighest weight L,+Ly+Ls.Theweights of\?V areoftwokinds: wehave theeight sums +L, +Ly+Ls,corresponding to thevertices ofthereference cube and each occurring once; and wehave the weights +L,each occurring twice (as+L,+L,—L; and +L,+Ly—Ly) ‘Theweight diagram thus looks likethevertices ofthereference cube together withthe midpoins ofitface 1% i ig ae Now, theweights +,must occur intherepresentation T,o,, with highest weightL,+Ly+Ls,incetheyarecongruenttoL,+Ly+Lymodulothe rootlatticeandlieintheconvexhullofthetranslatesofL,+Ly+Lsunder theWeylgroup(thatis,theylieintheclosedreference cube).But(heycannot occur with multiplicity greater than 1:forexample, theonly way togetfrom thepoint Ly+Ly+Lytothepoint L,bytranslations bythebasic vectorsaLusky~Lavand—2L,plturedinDiagram(17.1)abovewhilestayinginside thereference cube)isbytranslationby—2L,first,andthenbyLs—Ly. itfollows that themultiplicities oftheweights +L; inTo,o,; are1.Onthe other hand, wehave anatural map NVoV obtained bycontracting with theelement ofA?V* preserved bytheaction of, ‘sp6C, and thekernel ofthis map, which must contain therepresentation Trane illhave exactly these weights The kernel of ithus theirreducible representation with highest weight L,+Ly+Ly;wewilcallthisrepresenta- tion Ufor now ‘Atthis point, wehave established theexistence theorem forrepre- sentations ofspgC: theirreducible representation I,,.with highest weight (a+b +o)Ly +(a+b)L, +cLywilloccur inside therepresentation ‘Sym*V@Sym'W@Sym‘. $17.2. Representations ofsp inGeneral 289 Forexample,supposewewanttofindtheirreducible representation T,1,0‘withhighestweight2L,+L.Theweightsofthis representation willbethe 24weights +2L, +L,,each taken with multiplicity 1;the8weights +L, +L,£Ly,takenwithamultiplicity wedonotaprioriknow(butthatthereader can verify must beeither 1or2),and theweights +L, taken with someothermultiplicity. Atthesametime,therepresentation V®W,whichcontainsT,,1,0, wiltake onthese weights, with multiplicities 1,3,and6,respectively Inparticular, itfollows that V@Wwillcontain@copyoftheirreducible representation Uwith highest weight L,+L +Lsaswell, alternatively, we ‘canscethisdirectly byobserving thatthewedge product map V@NV>AV factors togive amap vewou andthatT),,9 must fieinthekernel ofthismap. Tosaymore about the location off;,,,0 inside V@W,anditsexact weights, would require eitherexplicitcalculation orsomething liketheWeylcharacterformula.Wewillsee inLecture 24how thelatter canbeused (osolve theproblem; forthetime being weleave thisas, Exercise 17.4. Verily bydirect calculationthatthemultiplicitiesoftheweights ofT,,s,0are 1,2,and 5,and hence that thekernelofthemapgaboveisexactly therepresentation I,0. §17.2. Representations ofsp;,€ inGeneral ‘Thegeneralpictureforrepresentations ofthesymplectic Liealgebrasoffersnofurther surprises. Aswehave seen, theweight lattice consists simply of integral linear combinations ofthe 1,And ourtypical Weyl chamber isacone ‘over asimplex inn-space, with edges therays defined by Oy=a=a>yyy===O.‘Theprimitivelatticeelementontheithrayistheweight=Ly+"+Lyandwemayobservethat,similarlytothecaseofthespeciallinearLiealgebras,these nfundamental weights generate asasemigroup theintersection of theclosed Weyl chamber with theJattice. Thus, our basic existence and ‘uniqueness theorem asserts that foranarbitrary n-tuple ofnatural numbers (ay, «5d,)€INthere willbeaunique irreducible representation with highest weight 1004 +304 +°°"+0 (ay HH a) (aybo ay)Lg +o +Ogle 260 17,66andsp ‘Asbefore, wedenote thisbyTy, Trrccata=Fattantlyoadtagll hg ‘These exhaust allirreducible representations ofsp24C. ‘Wecanfindtheirreducible representation V"=Tp,.s,...o With highest weight Ly++": +Lyeasily enough. Clearly, itwillbeconiained inthekth exterior power /Y ofthestandard representation. Moreover, wehave a natural contraction map oeNV NOY defined by airyAoAa)FOlOHY=WH AoAmeAGA ny (Gee§8.3 ofAppendix Bforanintrinsic definition andexplanation) Since the representation '-?V does nothave theweight L+" +La,theirreduciblerepresentation witthishighestweightwillhavetobecontainedinthekernelofthismap. Weclaim now that conversely ‘Theorem 175, For1<k<1the kernel ofthemap ois exactly theirreducible representation ¥™=T,.0.1,0,.,0 Withhighest weight Ly+°+" +Ly, Proor. Clearly, itisenough toshow that thekernel of@,isanirreduciblerepresentation ofsp,C.Wewilldothisbyrestrictingtoasubalgebra ofsp2,isomorphic tos,C, andusing what wehave learned about representationsof SC.TodescribethiscopyofslyCinsidespaxC,considerthesubgroupG<Spy oftransformations ofthespaceV=€2*preservingtheskewformQintroduced inLecture 16andpreserving aswellthedecomposition V=C{¢,..-s64) ® (eyes n4}- These canactarbitrarily onthefirst factor, aslong asthey dotheopposite onthesecond; incoordinates, they arethematrices x 0 Wehave, correspondingly, asubalgebra A 0 isomorphic tos1,€. Now, denote byWthestandard representation ofs1,C. The restriction oftherepresentation Vofepz4€tothesubalgebra sthensplits v=wewe intoadirect sum ofWanditsdual andwehave, correspondingly, $172, Representations ofsp;,€ inGeneral 261 AV= @uewenwe How does thetensor product "W@ A*W* decompose asarepresentation ‘ofs1,C7 Weknow theanswer tothis from thediscussion inLecture 15(see Exercise 15.30): wehave contraction maps Ye MW@ AW? howe AWs, and the kernel of¥,, isthe irreducible representation Wi!" =To0,400, With(i,say,<n—highestweight2L,+--"+2L,+ Lea +°F Ly-y. Therestriction ofA*Vtosisthusgivenby Ave wenok, andbythesame token, Ker) =@,we. Notethattheactualhighestweightfactorinthesummand W""&Ker()AV isthevector WOM meeyAo NEGA agents AoAae ENTREGA CagensantAOACaer Exercise 17.6. Show that more generally thehighest weight vector inany summand 1" cA‘V isthevector WhO me AT AOA Caectigss AOACay AEEN meA Ne,ACanctsnas AOACay (LUE AtgaEE Bytheabove, anysubspace ofKer(g,) invariant under sp2,C must bea irect sum, over asubset ofpairs (a,b) with a+b=k,ofsubspaces 1", Butnow (supposing forthemoment that k<n) weobserve that theelement Zanes=Eaecbia+Enten-b€PanC carties thevector w'" into w""!**” and, likewise, Yorncnes =Eavr.aener +Excnesarass ©8PanC carries w'* towi") Incasea+b=k=1,weseesimilarlythat Vom Eaten’ PC carries thevector w into w"!**, and Uses =Eartnsars ©9P20€ carries w"” tow'"*'#-Y, Thus, anyrepresentation ofsp2q€ contained in Ker(g,)andcontaininganyoneofthefactorsW‘*"willcontainthemall,and vweare done. o 262 17,5p6€ and spac Exercise 17.7.Another waytoconclude thisproof would betoremark that,inasmuchasallthew""aboveareeigenvectors ofdifferent weights, any highest weight vector fortheaction ofsp2,C onker(q) <A'Vwould havetobe(uptoscalars)oneofthew=".Itwouldthusbesufficienttofind,foreach (a,8)with a+b =k other than (a,b)= (k,0),apositive root asuch that .(w*") #0.Dothis. Note that, having found theirreducible representations VW"=To,...1...0 with highest weight Ly+" +Ly,anyother representation ofsp, will‘occurinatensorproductofthese;specifically, theirreducible representationTrccsong Withhighest weight ayLy4°°°-+ay(Ly +++L,)willoccur inthe product Sym*V@Sym**¥@--@Sym*V", ‘One further remark isthat there exist geometric interpretations ofthe action ofsl,,Conthefundamental representations V™.Wehavesaidbefore that thegroup PSp,,€ may becharacterized asthesubgroup ofPGL,C carrying isotropic subspaces ofVintoisotropic subspaces. Atthesame time,PGL«€actsontheprojective spaceP(/*V)astheconnected component oftheidentity inthegroup ofmotions ofthisspace carrying theGrassmannian G=G{k, V)cPUNY) intoitself. Now, thesubset G,<Gofk-dimensional isotropic subspaces ofVisexactly theintersection oftheGrassmannian G with thesubspace P(V®) associated tothekernel ofthemap @above; sothat PSpa4C willactonP(V™) carrying G,intoitself andindeed when 1<k<n ‘may becharacterized astheconnected component oftheidentity inthegroupofmotionsofP(V)preserving thevarietyG,. Exercise 17.8, Show that ifk>nthecontraction gis injective. §17.3. Weyl’s Construction forSymplectic Groups ‘Wehave just seen how thebasic representations forsp24C canbeobtainedbytakingcertainbasicrepresentations ofthelargerLiealgebrasl,,C—inthis case, MVfork<n—and intersecting with thekernel ofacontraction con: structed from thesymplectic form. Infact, alltherepresentations ofthe symplectic Liealgebras canbegiven asimilar conrete realization, byinter-sectingcertainoftheirreducible representations ofsl,withtheintersectionsofthekernels ofallsuch contractions. Recall from Lectures 6and15that theirreducible representations ofslaC aregiven bySchur functors $V, where 1=(A,2***2day2O)isa partitionofsomeintegerd=J:4,,andV=C™.Thisrepresentation isrealizedastheimage ofacorresponding Young symmetrizer ¢,acting onthed-fold tensor product space V®, Foreach pair!={p<q) ofintegers between |andd,thesymplectic form@determines acontraction 0:V8youn, (179) 2B BoyOp0—)2)BOI, OBO BWve 4173.Weyl'sConstruction forSymplectiGroups 263 LetV&¥®denotetheintersection ofthekernelsofallthesecontractions. ‘Thesesubspacesismappedtoitselfbypermutations, soVisasubrepresen- tation ofV®asarepresentation ofthe symmetric group ,.Now let" SaV=VOOS,¥. (17.10) ‘This space is@representation ofthe symplectic group Spz4C ofQ,since V and5,(V) aresubrepresentations ofV%over Sp,C. ‘Theorem17.11.ThespaceSy,(V)isnonzeroifandonlyiftheYoungdiagramofBhas atmost nrows, Le,Ayss=0.Imthiscase, Say(V) istheirreduciblerepresentation of84withhighestweight2yLy4°"2gby. Inother words, foranntuple (a,,..-»a,) ofnon-negative integers Taste =Says where 1sthepartition (ay+ay4°" +0q)024 °°"dyy ony)Theprooffollowsthepatternforthegenerallineargroupgivenin§6.2,but wewillhave tocallonabasic result from invariant theory inplace ofthe simple Lemma 6.23. Wefistshow how tofindacomplement toV inV4 For example, ifd=2,then V9 VOC, where istheelement ofV@ Vcorresponding tothequadratic form Q.In terms ofourcanonical basis,y=¥(¢,@e41—eno:@€;).Ingeneral,forany I=(p<4} define yyvee. you byinserting yinthep,q factors. Note that ©,0, ismultiplication by 2n=dim VonV4", We claim that VemVOOYHVE), (17.12) Toprove ths, putthestandard Hermitian metric, Jon V=€%, using thesiven¢,aabasis,sothat(ae,be,)=dab.ThisextendstogiveaHermitianmetric oneach V®. Weclaim thatthedisplayed equation isaperpendicular direct sum. This follows from thefollowing exercise. Exercise 17.13. (i)Verify that for6,we¥;(¥-0@»)=Ole.w) {)Use(9)toshow thatKer(@,) =Im('¥,} foreach 7 Now define F< V® tobetheintersection ofthekernels ofallr-fold contractions ®,,0-0 @,,,and set V68, =I), 0°8,VEE™), (1714) "This flows aclassical notation ofwing ¢forthe symplectic group and{forthecxinogonagroup(althoughwehaveomitedthecorespnding notation{)forthegeneral Bacar oun) 24 17.sp6€ and spn Lemma17.18.ThetensorpowerV®decomposes intoadirectsum VO=VODVROVRO OVApy with p=(4/2), and,forallr>1, FE VO OVO” OV Exercise 17.16. (i)Show asinthepreceding exercise that there isaperpendi- cular decomposition VOE=FA@YYH)07-0,(VON). (ii)Verity that(F$7)<Fay. (ii)Show byinduction thatV®isthesumofthespaces V2, (iv)Finish theproof ofthelemma, using (i)and(ji)todeduce thatboth sums areorthogonal splittings. o Allthesubspaces inthese splittings areinvariant bytheaction ofthe symplectic group Sp2,C, aswell astheaction ofthesymmetric group S,.In particular, weseethat Sash=VOrey=ImleyVO+V®) 77) Exercise 17.18%. ()Show that ifs>n,then M'V@V@*-"iscontainedin Lr¥AVE"), anddeduce that $4)(V) =OilAyu isnot0. {i}Show thatS.4,(V)isnotzeroif2,4,=0. Foranypair ofintegersIfrom(1,.-.d),define9,=p0.0:VOVE, From what wehave seen, 1 isthe intersection ofthe kernels ofallthese endomorphisms. Note that theendomorphism ofV®determined byany symplectic automorphism ofVnotonly commutes with allpermutations of thefactors &,butalsocommutes with theoperators 9,.Weneed afactwhich isproved inAppendix F.2: Invariant Theory Fact 17.19. Any endomorphism ofV® that commutes with allpermutations inS,andalltheoperators 9,isafinite C-inear combinationOfoperatorsoftheformA@-""®A,forA€Sp, Now letBbethealgebra ofallendomorphisms ofthespace VthatareC-linearcombinations ofoperatorsoftheformA@--:@A,forAeSpasC- Proposition 17.20. Thealgebra Bisprecisely thealgebra ofall endomorphisms ofVcommuting withallpernaitations inS. Proor. IfFisanendomorphism ofVcommuting withallpermutations of +seat than the andamarnhiem FofV% that isFonthe factor V and §173. Weyl Construction forSymplecic Groups 265 zeroonthecomplementary summand J.,¥,(¥#") isanendomorphism that commutes with allpermutations andalloperators 9).The factthat F isalinear combination ofoperators from thesymplectic group (which we ‘know from Fact 17.19) implies thesame forF. o Corollary 17.21. Therepresentations S.,(V)areirreduciblerepresentations of Spi,C Proor. Since Bisthecommutator algebra toA=C[S,] acting onthespace V, Lemma 6.22implies that(V)-c, isan irreducible B-module. Butwe have seen that(V‘®)-c, =Sci)¥, and theproposition shows that being irreducible over Bisthesame asbeing irreducible over Spa,C. o Exercise17.22.Showthatthemultiplicity withwhichS.,,(V)occursin¥isthedimension m,ofthe corresponding representation V,ofSy ‘Aswas thecase fortheWeyl construction over GLC, there aregeneralformulasfordecomposing tensorproductsofthese representations, aswell as restrictions tosubgroups Sp,,-€, and fortheir dimensions and multiplicities ofweight spaces. Wepostpone these questions toLecture 25,when wewill have theWeyl character formula atourdisposal. [AswesawinLecture 15forGLC; itispossibie tomake acommutativealgebrawhichwedenotebyS©=$(1)outofthesumofall theirreducible representations ofSp.,€,whereV=C™isthestandardrepresentation. Prob-ably thesimplest waytodothis,given what wehave proved sofa,isto start with thering AV,n) =SymVONVONVO-ONV) =D, Sym(NV) @-~- @Syme) @Syme(V), thesum over alln-tuples a=(a,,...,4,) ofnon-negative integers. Define aring$'(V,n)tobethequotientof4°,n)bytheidealgeneratedbythesamerelations asin(15.53). Bytheargument in§15.5, thering $'(V,n)is thedirect ‘sum ofall therepresentations $,(V) ofGL(V}, a2varies over allpartitions with atmost nparts. ‘Thedecomposition V4=V®@W of(17.12) determines adecom- position Vc, =Vc, @Wc, which isadecomposition SAV) =Sa(M Olav) ofrepresentations ofSp,,€.WeclaimthatthesumJ=@,Jcy(V) isanidealinS'(V,n) =@,Sil). Thisiseasytosceusing weights, since Jc,,(V) isthesum ofall therepresentations in$,(V) whose highest weight isstrictly smaller than 2.This implies thattheimage ofJ,,,(V)® 8,(V) inS4,4(V) is 4sum ofrepresentations whose highest weights arelessthan 2+ $0they aust bein Jea.(V). Thequotient ringi,therefore, thering $°(V)wewerelookingfor: 266 17,sp6CandpC 50=SUM =PSa) Infact, theidealJ“isgenerated byelementsofthe form x»y,wherex€N'V, i-<n—2,and /istheelement inA*V corresponding totheskew form Q.An ‘oulline oftheproofissketchedattheendofLecture25.Thecalculations, as wellasotherconstructions ofthe ring, canbefound in[L-T], where onecan also find adiscussion offunctorial properties oftheconstruction. For bases,see(DC-P],[L-M-S], and[M-S] LECTURE 18 Orthogonal LieAlgebras Inthisand thefollowing tw. lectures wecary outforthe orthogonal Liealgebraswhatwehavealreadydoneithespeciallinearandsymplecticeases.Asinthoseeases,westart byworking outivgeneral thestructure oftheorthogonal Liealgebras, Gescribing theroots, root spaces, Weyl group, etc, and then gotowork onlow dimensional examples. Therefsnenewpenotenon her:asitturnsoutallthreof theLiealgebraswedealwithin§18,2areisomorphic tosymplecticorspeciallinear Liealgebras wehave already analyzed (this willbetrue ofsogC aswell, butofnoother orthogonal Lialgebra) Asinthe previous cases, theanalysis ofthe Lealgebras and their representation theory willbecompletely elementary. Algebraic gcometty doesintrudeintothediscussion,howeverwehavedescribedtheisomorphism betweentheorthogonal Liealgebras discuseed and special linear and symplectic ones interms of Projectve geometry since that iswhal seems tous most natural. This should notbeAproblem;thereacemanyotherwayeofdescrbingtheseomorphioms, andreaderswho disugree with our choice ean substitute their own, $18.1: $0,C and2046 18.2; Representations of¢93C, e04C, and 604 §18.1. SO,,C and s0,,C Wewilltake upnow theanalysis oftheLicalgebras oforthogonal groups. Here there is.aswewillseevery shortly, »very bigdifference inbehavior between theso-called “even” orthogonal LicalycLias $0,,€ and the“odd”orthogonal Liealgebras+0,,,,. Interestingly enouyh,thelatterseematfirst glance tobemore coipiicated, especially interms ofnotation; butwhen we analyze their representations wesec “hatinfact thybehave more regularly ‘than theeven ones. Inanyevent,wewlltrytocarryouttheanalysisinparallel 268 18,OrthogonalLieAlgebras fashion foraslong asisfeasible; when itbecomes necessary tosplit upinto ‘cases, wewillusually look attheeven orthogonal Liealgebras first and then consider the odd, LetVbeam-dimensional complex vector space, and O:VxV4e ‘anondegenerate, symmetric bilinear forin onV.The orthogonal group SO,C isthen defined tobethegroup ofautomorphisms AofVofdeterminant 1 preserving Q—that is,suchthatQ(Av,Aw)=Q(v,w)forallv,weV—and the orthogonal Liealgebra so,,C correspondingly consists ofendomorphisms A:VV satistying Q(Av, w)+Qto, Aw)=0 (18.1) forallvand weV.Asinthecase ofthesymplectic Liealgebras, tocarry out‘ouranalysiswewanttowriteQexplicitly intermsofabasisforV,andhereiswherethecasesofevenandoddmfirstseparate. Incasem=2niseven,wewillchoose abasisforVintermsofwhichthequadratic formQisgivenby lei sn) =Oleisw e)=1 and Ole,e)=0 ifj#itn. Thebilinear form Qmay beexpressed as Ole») = Moy, whereMisthe2nx2nmatrixgiveninblockformas 01 Me(,) thegroup SO;,C isthus thegroup of2nx2nmatrices Asatisfying M='A-M’A andtheLiealgebra s0,,€ correspondingly thespace ofmatrices Xsatisfying the relation 'X°M +M-X =0. ‘Writing a2nx2nmatrix Xinblock form as AB xe(2) we have ‘CA XM =w-(54) and $18.1. SO,C and50, 269 cD) sothat thisrelation isequivalent tosaying that theoff-diagonal blocks Band CofXareskew-symmetric, andthediagonal blocks AandDofXarenegative transposes ofeach other. Exercise 18.2, Show that with this choice ofbasis, a0 .s,0-{(0 2)}se. and s0,€ =. ThesituationincasethedimensionmofVisoddissimilar,ifalittlemessier. Tobeginwith,wewilltakeQtobeexpressible, intermsofabasis€,,..-s€an41 for¥,by Qe esa) =Deiter) =1 fort sis; Qlernst Czas)=5 and Qle,,€)) =0forallother pairs i,j ThebilinearformQmaybeexpressed as Q(x, y)=e Moy, whereMisthe(2n+1)x(2n-+1)matrix a M={1,[0[0 otolt (thediagonalblocksherehavingwidthsn,n,and1).TheLiealgebras02—4;Ciscorrespondingly thespaceofmatricesXsatisfying therelation‘XM+M-X =0; ifwewrite Xinblock form as A|BIE x=| c[p|F |, Grails then this isequivalent tosaying that, asintheprevious case, Band Care skew-symmetric andAandDnegative transposes ofeach other; andinaddition E=-'H,F =~'G,andJ=0.With these choices, wemay take asCartan subalg-bra—in both theeven andoddcases—the subalgebra ofmatrices diagonal inthisrepresentation." "Notethatifwehadtakenthesimplerchooeof©,withMtheidentitymarx,theLiealgebrawould have consisted ofskew-symmetric matrices, and there would have been nononzero diagonal matrices inthe Lie alecbva 270 18,Orthogonal LieAlgebras The subalgebra fisthus generated bythen2nx2nmatrices Hy=E,— Egsinvi Whose action onVisto fixe,,send e,,,toitsnegative, andKillallthe remaining basis vectors; note that thisisthesame whether m=2nor2n+1. Wewillcorrespondingly take asbasis forthedual vector space *thedual basis L),where <L),H,)=5,GiventhattheCartansubalgebra ofs0,,€coincides, asasubspaceofs12,€, with theCartan subalgebra ofsp,,C, wecanusemuch ofthedescription of theroots ofsp2,€ tohelp locate theroots and root spaces ofs0,4C. For example, wesaw inLecture 16that theendomorphism X= Ej Entiat ©P20 isaneigenvector fortheaction ofhwitheigenvalueL,—L,.SinceX,,isalso anelement ofs0,4€;, weseethatL,—Lyslikewise arootof$024, withroot space generated byX,,.Lessdirectly butusing thesame analysis, wefindthat theendomorphisms Yi=Fonts —Enns and 25=Enis Entit areeigenvectors fortheaction of,witheigenvalues Ly+L,and—Ly—Ly respectively (note that ¥,,andZ,,,donotcoincide with their definitions in Lecture 16).Insum, then, therootsoftheLiealgebras0,,€arethevectors {thi tLhe b* ‘Thecase ofthealgebra 0344, issimilar; indeed, i!theeigenvectors for theaction ofhfound above ins0,,€, viewed asendomorphisms of€2*"", are likewise eigenvectors fortheaction ofon02,4, €.inaddition, wehave the endomorphisms Up=Exansr ~Exesinti and Ve=Enstansr ~Exesswhichareeigenvectors witheigenvalues +L,and—L,,respectively. TherootsOf802941€ arethus theroots +L;+L;of$029, together with additional roots +L; Wenotethatwecouldhavearrivedatthesestatementswithoutdecompos- ingtheLiealgebrass0,.C:thedescription (18.1)oftheorthogonal Liealgebramay beinterpreted assaying that, interms oftheidentification ofVwith V* given bytheform Q,sog€ isjusttheLiealgebra ofskew-symmetric enclo- morphisms ofV(anendomorphism being skew-symmetric ifitis equal to‘minusitstranspose). Thatis,theadjointrepresentation vf50,isisomorphictothewedge product (PV. Intheeven casem=2n,since theweights ofVare +L,(inasmuch asthesubalgebras h<End(V) coincide, theweights ofVmustlikewisebethesamefors0,,Casforsp2,C),itfollowsthattherootsofs0,C $18.1, $O,Cands0,€ m arejustthepairwise distinct sums +L,+LyIntheoddcasem=2n+1,we seethat e245: €Visaneigenvector fortheaction of§with eigenvalue 0,so that theweights ofthestandard representation Vare{+L;} U{0}andthe weights oftheadjoint representation correspondingly (4L;+Lj}U{+Ly}. Exercise18.3,Useasimilaranalysistofindtherootsofsp,Cwithoutexplicitcalculation. Tomakeacompisison withtheLiealgebrasp,,C,wecansaythattheroot diagram of802.€ looks like that ofsp24€ with theroots +2L, removed, ‘whereas theroot diagram of503441 looks likethat ofsp,4 with theroots £2L, replaced by+L;. Note that this immediately tells uswhat theWeyl ‘BrOUpS are:first, inthecase of50,4, theWeyl group isthesame asthat of SP2gC! 14ZY +B, Sh Inthecase of$02,C, theWeyl group isthesubgroup oftheWeyl group of sP2,€ generated byreflection inthehyperplanes perpendicular totheroots+L;+Ly,withouttheadditional generator givenbyreflection intheroots£L,. This subgroup still acts asthefullsymmetric group onthesetofcoordinate axesin5*;butthekernelofthisaction,insteadofactingas++oneach ofthecoordinate axes independently, willconsist oftransformations ofdeterminant 1;ie,willactas—1onanevennumberofaxes.(Thateverysuch transformation isindeed intheWeyl group iseasy tosee: forexample, reflection intheplane perpendicular toL,+1,;followed byreflection inthe plane perpendicular toL,—L;willsend L;to Lj,Ljto~Lj, andLytoLy fork#i,j.)Another way tosaythisisthat theWeyl group isthesubgroupoftheWeylgroupofsp,consistingoftransformations whosedeterminant agrees with thesign oftheinduced permutation ofthecoordinate axes; sothat while theWeyl group ofspz4C fitsinto theexact sequence 1-+(Z/2) +BWyy,6+ Se 1, theWeyl group ofs0,4C hasinstead thesequence 14(Z2Y"! +By,,,0-+ Se-¥ Wecan likewise describe theWeyl chambers ofs0,,€ and 603,4,€ by direct comparison with sp,,C. Tostart, tochoose anordering oftheroots Wetake aslinear functional onb*aform [=cH, +--+ CylHly, where 6>¢3>"+">>0.Thepositiverootsinthecaseof$02.4,€arethen R= (byt high {Lim Lijicit Lhe whereas inthecase of50,4 wehave RY={Ly+Lhd (Li Lies The primitive positive roots are m 18,Orthogonal LieAlgebras Dy=yyLy—Layeonybyes=byby for$0446 Ly=bgyby—bgpooeyLye ~byLynn +Lyf0r6034C. Inthefirst case, theWeyl chamber isexactly thesame asforsp,,C, namely, form=2n+1, W=(Labya, 2a; >" 24,>0} since theroots arethesame except forthefactor of2onsome.Inthecaseof '80,C, since there isnoroot along theline spanned bytheL,,theequality a,=0does notdescribe afaceoftheWeylchamber; however, sinceL,_,+Ly isstillaroot(and apositive one) westillhave theinequality a,-, +a,>0in W,sothat wecanwrite, form=2n, W={Sabi a,>a,>"+2ay>1a}. (Note thatinthecase ofs0,€ wecould have chosen ourlinear functional LacyHy +0 +eH, with cy>c; >"> —c,>0; theordering ofthe roots, andconsequently theWeyl chamber, would stillbethesame.) AsfortheKilling form, thesame considerations asforthesymplectic case show thatitmust be,uptoscalars, thestandard quadratic form: BH, H,)=6,,.(ThiswasimplicitintheabovedescriptionoftheWeylgroup.)Theexplicit calculation isnomore difficult, and weleave itasanexercise: ((4n—2)S ab,ifm=2n+1 BYaH,YH)={im~4)Eqh, itm=2n Next, todescribe therepresentations oftheorthogonal Liealgebras we have todetermine theweight lattice inh*;andtodothiswemust, asbefore, locate thecopies s,ofst,€ corresponding totheroot pairs +a, and the corresponding distinguished elements H,ofh.This issosimilar tothecaseof ‘5P2aC that wewillleave theactual calculations asanexercise; wewillsimply state here theresults that ins0,,€ foranym, {i)thedistinguished copy s,,-,, ofslzC associated totherootL,~Ly isthespan oftheroot spaces q,,-1, =C°X),j 9-1-1, =©°Xj,andtheir commutator [X;,,Xi]=Eyy~Ey.j+Easiness~Envicnsir Withdistinguishedelement H,-1, =H;—H,(thisisexactly asinthecaseofsp,_C); (ii)thedistinguished copy s,,42, ofsl, associated totheroot L,+Ly isthespan oftherootspaces fis, =" ¥.yQ-t-1, =C*Z,, andtheir commutator [¥.p Zij) =—Bist Ey Basins +Enstant =—Hi~ Hy withdistinguished element H,,,., =Hy+H,(60thatwehavealsoH_y,-., =—H,—Hj);andinthecaseof803,,,C,(iii)thedistinguished copy,,ofsl,associatedtotherootL,isthespan oftherootspaces q,,=€-U,,g-.,=€-¥; andtheircommutator [U,,4]= CELants —Exettnsis Entiantt ~Exnec) =—Howith distinguished element H,,=2H,(sothatH_,, =—2H, aswell). Exercise 18.4, Verify thecomputations made here. $182. Representations of90,0,€andsos 23 Again,theconfiguration ofdistinguished elements resembles that ofsp, closely; that ofs0.,41€ differs from itbythesubstitui‘on of+24, for+H, whereas that ofs03,€ differs bytheremoval ofthe4:H. The effect onthe Weight lattice isthesame ineither case: forboth even andoddorthogonal Lie algebras, theweight lattice Ayisthelattice generated bytheL,together withtheelement(Ly+°°"+L,)/2. Exercise 18.5. Show that zp ifm=2n+1 Aw/Mn= 4Z/4 ifm=2nandnisodd (2227/2 ifm=2mand niseven. §18.2. Representations ofso, so,C, audso,C Togive some examples, start with thecase n=1.Ofcourse, s0,€ =Cisnotsemisimple, Therootsystemof50,ontheotherhand,lookslikethatofst,C: —e——_} by ° Bo. This isbecause, infact, thetwo Liealgebras areisomorphic. Indeed, likethe symplectic group, thequotient PSO,€ oftheorthogonal group byitscenter canberealized asthemotions oftheprojective space PVpreserving isotropic subspaces forthequadratic form Q;inparticular, this means wecan real- izePSO,C asthegroup ofmotions ofPY=P"? carrying thequadric hypersurface @={Lv}: Qe,v)=0} intoitself. Inthefirstcase ofthis, weseethatthegroup PSO, isthegroup‘ofmotionsoftheprojective planeP?carryingaconiccurveC<P?intoitself Butwehave seen before that thisgroup isalso PGL3C (theconic curve isitselfisomorphic toP!,andthegroupactsasitsfullgroupofautomorphisms),givingustheisomorphism 50,~sl,C.Onethingtonotehereisthatthe“standard” representation ofs04C isnotthestandard representation ofst, butrather itssymmetric square. Infact, theirreducible representation withhhighestweight}L,isnotcontained intensorpowersofthestandardrepresentationofoC.Thiswillturnouttobesignificant: thestandardrepresentationofsl,€, viewed asarepresentation ofs05€, isthefirst example ofaspin representation ofanorthogonal Liealgebra.Thenextexamplesinvolvetwo-dimensional Cartanalgebras.Firstwehaveso4C, whose root diagram looks like 24 18,Orthogonal LieAlgebras itesx|tthe whyDX fits Note onething about thisdiagram: theroots arelocated ontheunion of twocomplementary lines. This says, byExercise 14.33, that theLiealgebra 504 isdecomposable, andinfactshould bethesum oftwoalgebras each of whose root dingrams looks ikethatofsC; explicitly, s04C isthedirect sumofthetwoalgebrass,,fora=Ly+Lzanda=Ly~La.Infact,wecansee thisisomorphism s04€ &sl€xsh3C, (186) asintheprevious example, geometrically. Precisely, wemay realize thegroup PSO,C =SO,C/{ £1}astheconnected component oftheidentity inthe group ofmotions ofprojective three-space P?carrying aquadric hyper- surface Qintoitself, Butaquadric hypersurface inP*hastworulings byTines,andthesetworulingsgiveanisomorphism of@withaproductP!xP" PSO,CthusactsontheproductP!xP!;andsincetheconnected componentoftheidentity intheautomorphism group ofthisvariety isjusttheproduct PGL,C xPGL,C, wegetaninclusion PSO,C-+ PGL,€ xPGLC. Another way ofsaying thisistoremark that PSO,C acts onthevariety of isotropic 2-planes forthequadratic form QonV;andthisvariety isjustthe disjoint union oftwocopies ofP!.Toseeinthiscase that themap isan §18.2. Representations of0,C, s0,€, and sos 28 isomorphism, consider thetensor product V=U@Wofthepullbacks tosl,Cxsl,Cofthe standard representations ofthetwofactors. Clearly the action onP(U ®W)willpreserve thepoints corresponding todecomposable tensors (that is,points oftheform [u@ w])}; butthelocus ofsuch points is justaquadric hypersurface, giving ustheinverse inclusion ofPGL,C x PGL,C inPSO,C. Tnfact,allofthiswillfalloutoftheanalysisoftherepresentations ofso,C, iffwejust pursue itasusual.Tobeginwith,theWeylchamberwehaveselected looks like . » fae, Now, thestandard representation has, asnoted above, weight diagram >maar‘ withhighest weightL,(notethatthehighest weightofthe standard represen- tation liesinthiscase intheinterior oftheWeyl chamber, somethirig ofan anomaly). Itssecond exterior power will have weights +L; +1. and 0 (occurring with multiplicity 2),ie,diagram 216 18.Orthogonal LieAlgebras Weseeonethingaboutthisrepresentation rightaway,namely,thatitcannotbeirreducible. Indeed, theimages ofthehighest weight L,+Lyunder the Weyl group consist just of(Ly +La),80thatthediagramoftheirreducible representation with thishighest weight is Weseefrom thisthatthesecond exterior power AV ofthestandardrepresentation ofs04€mustbethedirectsumofthe irreducible represen- tations W,=i, and W,=Ty,-1, with highest weights L,+L, and L,—Ly.Since A*V isatthesame time theadjoint representation, thissaysthat60,CitselfmustbeaproductofLiealgebraswithadjointrepresentations Taye, andly-1,)‘OnewaytodetivethepictureoftherulingofthequadricinP?fromthis,decomposition istoview so,€ asasubalgebra ofslyC, and theaction of PSO,C onP(A?V) asasubgroup ofthegroup ofmotions ofP2(\7V) =PS preserving theGrassmannian G=G(2, V)oflines inP%,Infact, weseefromtheabovethattheactionofPSO,onP*will preserveapairofcomplementary 2-planes PW, andPW,,; itfollows thatthisaction must carry into themselves $18.2. Representations ofs05€, 8040, and605 27 theintersections ofthese2-planeswiththeGrassmannian. Theseintersections areconic curves, corresponding toone-parameter families oflines sweeping outaquadric surface (necessarily thesame quadric, since theaction ofSO,Conpreserves uniquequadraticform;thus,thetworulingsofthequadric. Pw Pw; _G=cam Note onemore aspect ofthisexample: asinthecase ofs0,€ %slyC, theweightsofthestandard representation ofs0,€donotgeneratetheweightlattice,butratherasublattice Z{L,,L,}ofindex 2inAy. Thus, there isno wayofconstructing alltherepresentations ofso, byapplying lineat- or multilinear-algebraic constructions tothestandard representation; itisonly aller weareaware oftheisomorphism so, %sl;C xslC that wecan construct, forexample, therepresentation Ty,+1,y2 with highest weight (Ly+L,)/2 (ofcourse, this isjust thepullback from thefirst factor of sl€ xslyC ofthestandard representati-n ofs1,€). ‘Wecome now tothecase ofs05€, which ismore interesting. The root diagram inthis case looks like xt 278 1%,Orthogonal LieAlgebras {asinthepreceding example, theweight lattice isthelattice ofintersectionsofallthelinesdrawn).Thefirstthingweshouldnoticeaboutthisdiagramisthat itisisomorphic totheweight diagram oftheLiealgebra sp4C; the diagram justappears here rotated through anangleof7/4.Indeed,thisisnot accidental; thetwo Liealgebras sp. and s04C areisomorphic, and itis riothard {0construct this isomorphism explicitly. Toseetheisomorphism geometrically, wesimply have torecall theidentification, made inLecture 14,ofthegroup PSp,C with agroup ofmotions ofP*,There, wesaw that thelarger group PGL4C could beidentified with theautomorphisms oftheprojective space P(A?V) =P*preserving theGrassmannian G= GQ,4)<PUY). Thesubgroup PSp,C©PGL,Cthuspreservesboththe GrassmannianG,whichisaquadcichypersurface inPS,andthedecompos- tionofAVintothespanCQoftheskewformQe\?¥*=A'Vanditscomplement 1,andsoactsonPIVcarrying theintersection G,=GnPW intoitself. Wethus sawthat PSp,C wasasubgroup ofthegroup ofmotions ofprojective space P*preserving aquadric hypersurface, and asserted that in factitwas thewhole group. (To seethereverse inclusion directly, wecan invoke alittle algebraic geometry, which tellsusthatthelocus ofisotropic linesforaquadcic inP*is isomorphic toP?,so thatPSOsC actsonP?,Moreover, thisaction preserves thesubset ofpairs ofpoints inP?whose corresponding linesinP*intersect, which, forasuitably defined skew-symmetric bilinear form 0,isexactly the setofpairs(Lv],[W])suchthatO(»,w)=0,sothatwehaveaninclusion of PSO,C inPSp,C) Let usproceed toanalyze the: representations ofsos aswewould ordinarily, bearing inmind theisomorphism with sp,C. Tobegin with, we draw theWeyl chamber picked outabove in Asfortherepresentations ofso,C, wehave (obegin with thestandard, which hasweight diagram $182. Representations of505€, s04€, and805 219 be Thisweseecorresponds totherepresentation W=A?V/C-Q ofsp,€. Next, thesecond exterior power ofthestandard representation ofso, hasweights peehy This isofcourse theadjoint representation ofsosC; itistheirreducible representation with highest weight L+Ly.Note that itcorresponds tothe symmetric square Sym*Vofthestandardrepresentation ofsp, (seeExercise 16.8). Exercise 18.7. Show thatcontraction with thequadratic form Q¢Sym*V* preserved bytheaction ofsos induces maps ‘9:Sym*V +Sym*"?V, ‘Show that thekernel ofthisCoustraction isexactly theirreducible representa- tion with highest weight a-L,. Compare thiswith theanalysis inExercise 1611 280 18,Orthogonal LieAlgebras Exercise18.8.Examinethesymmetric powerSym*(/\?V) oftherepresentation/2V.This willcontain acopy oftheirreducible representation Pyu,+14j; Whatelsewillitcontain?Interprettheseotherfactorsinlightofthe isomorphism £05€ =p,C. Exercise 18,9, For anexample ofa“mixed” tensor, consider theirreduciblerepresentation Ty,,1,-Showthatthisiscontained inthekernelsofthewedgeproduct map. oOVONV AV and thecomposition O:VONV+VE@NV SY, wherethefirstmapisinduced bytheisomorphism 0:V+V*andthesecond isthecontraction V*@A? —V.Isitequal totheintersection ofthese kernels? Show thattheweight diagram ofthisrepresentation is SRMIRS After youaredone with thisanalysis, compare with theanalysis given ofthe corresponding representation inLecture 16. Note that, asinthecase oftheother orthogonal Liealgebras studied so far(and asisthecase foralls0,,C), theweights ofthestandard representation donot generare theweight lattice, butonlythesublatticeofindextwogenerated bytheL,.Ths, thetensor algebra ofthestandard representation willcontain only one-halfofalltheirreduciblerepresentations ofso,€.Now,wedoknow that there areothers, andeven something about them-—for example, wesee inthefollowing exercise that theirreducible representation ofsos withhighestweight(Ly+L3)/2is.sortof"symmetric squareroot”ofthe adjoint representation: Exercise 18.10. Show, using only root andweight diagrams forsosC, thatthe ‘exterior square /?Vofthestandard representation ofsosC isactually the syuimetric square ofanirreducible representation $182,Representations of50,C,504C,andsos 281 ‘Wecanalso describe thisirreducible representation viatheisomorphism ofsosC withsp,C: itisjustthestandard representation ofsp,C onC*.We donotatthispoint have, however, away ofconstructing thisrepresentation without invoking theisomorphism. This representation, therepresentation of0€withhighestweightL1/2,andtherepresentation ofs04€withhighestweight (L,+L3)/2 discussed above arecalled spin representations ofthe corresponding Liealgebras and willbethesubject matter ofLecture 20. LECTURE 19 80,C, $0,C, andso,,C ‘This lecture isanalogous incontent (and prerequisites) toLecture 17:wedosome morelow-dimensional examples andthendescribe thegeneral pictureoftherepresen-tations oftheorthogonal Liealgebras. One difference isthatonly halftheirreducible representationsofs0,Clienthetensoralgcbraofthestandard,tocompletethepicture ofthe representation theory wehave toconstruct thespin representations, which is thesubject matter ofthefollowing lecture, The first four sections arecompletelyelementary (exceptpossiblyforthediscusionoftheisomorphism sagC>sl4Cin5191); thelatsection assumesaknowledgeofLecture6and§15.3,butcanbeskipped bythosewhodidnotreadthosesections. $19.1:Representations ofso,C §192: Representations oftheeven orthogonal algebras $19.3: Representationsofso, §19.4:Representations oftheoddorthogonal algebras $1955: Wey's construction fororthogonal groups §19.1. Representations ofso,C Wecontinue ourdiscussion oforthogonal Liealgebras with theexample of s0¢C. First, itsroot diagram: §19.1. Representations ofso,C 283 batty re 1 Lythyle ¢ byt 1 L * ep-Ly ‘Once more (and forthelasttime), wenotice acoincidence between this and theroot diagram ofaLiealgebra already studied, namely, sl,C. Infact, the twoLiealgebras areisomorphic. Theisomorphism isonewehave already observed, inasense: inthepreceding lecture wenoted that ifVisafour- dimensional vector space, then thegroup PGL,€ may berealized astheconnected component oftheidentityinthegroupofmotionsofP(\*V)=BS carrying theGrassmannian G=G(2,4)<P(A*V) intoitself, andPSpyC < PGL,€ thesubgroup fixing ahyperplane PW=P*cPS.Weused thisto identify thesubgroup PSp,C with theorthogonal group PSO,C; atthesame time itgives anidentification ofthelargergroupPGL,C withtheorthogonal group PSO,C. Even though so¢C isisomorphic toaLiealgebra wehave already examined, itisworthgoingthroughtheanalysisofitsrepresentations forwhatamountstoasecond time, partly soastounderstand theisomorphism better, but mainly because wewillseeclearly inthecase ofsog€ anumber ofphenomena thatwillhold true oftheeven orthogonal groups ingeneral. Tostart, wedraw theWeyl chamber inh*: 1S 1 mob Labastaty Asusual, webegin with thestandard representation, which hasweights +L,, corresponding tothecenters ofthefaces ofthecube: 284 19.806€, 60,6, and60,C iabs mFaas i ee ret el “7 * ‘NotethatthehighestweightL,oncemoreliesonanedgeoftheWey!chamber(the front edge, inthediagram onthepreceding page). Observe that the standard representation ofsogC corresponds,aswehavealreadypointedout, totheexterior square ofthestandard representation ofs1,C. Next,welookattheexteriorsquare/\?Vofthestandardrepresentation ofsogC.Thiswillhaveweights +L;+L,(ofcourse,itistieadjointrepresen- tation) and sowillhave weight diagram. <a 4 oon./LAN ‘Note thatthehighest weight vector L,+1..ofthisrepresentation does not lieonanedge oftheWeyl chamber, butrather intheinterior ofaface (the ‘back face, inthediagram above). Inorder togenerate alltherepresentations, ‘westillneed tofind theirreducible representations with highest weight along, theremaining twoedges oftheWeyl chamber. Welook next attheexterior cube /\*V ofthestandard representation. The weights here aretheeight weights +L, +L,+Ly,each taken with multi- plicity one, andthesixweights +L,,each taken with multiplicity 2,asinthe diagram ig tetas ' |© a ®| |® ® ' eee eee Cree x ® §19.1. Representations ofsog 285 Now, wenotice something very interesting: this cannot beanirreduciblerepresentation, Wecanseethisinanumberofways:theimagesoftheweightL,+Lz+LyundertheWeylgroup,forexample,consistofeveryothervertexOfthereference cube; inparticular, their convex hull does notcontain theremaining fourverticesincludingL,+Ly—Ly.Equivalently, thereisnowaytogofrom L,+Ly+LytoLy+Ly—Lybytranslation bynegative root vectors. Therepresentation /°Vwillthuscontain copies oftheirreduciblerepresentations Ty,41,41,aNdTr,1,-1,WithhighestweightsL,+Ly+Lsand Ly+Ly—Ls,with weight diagiams Lytegtey “AT s NG by AEN ee and a Lytleby Since theweight diagram ofeach ofthese isatetrahedron containing theweights+tL;,wehaveaccounted foralltheweightsof(Vandsomusthaveadirect sum decomposition OV=Vigetyets®Taystanny Wecanrelate thisdirect sum decomposition toageometric feature ofa ‘quadric hypersurface inPS,analogous tothepresence oftworulings ona ‘quadric inP?,Wesawbefore thatthelocus oflines lying onaquadric surface inP?turns outtobedisconnected, consisting oftwocomponents 286 19.s96€, s05€, and60,€ each isomorphic toP'(and embedded, viathePliicker embedding ofthe Grassmannian G=G(2,4)oflines inP®inP(\?C) =P*,asapairofconicccurveslyingincomplementary 2-planesinP5),Inasimilarfashion,thevariety of2-planeslyingonaquadrichypersurface inPturnsouttobedisconnected,consisting oftwocomponents that, under thePliicker embedding ofG(3,6) inP(\C*) =P'?,span twocomplementary 9-planes PW, andPW,; these ‘twoplanes givethedirect sumdecomposition of*Vasans0,C-module, Infact,ifwethink ofaquadric hypersurface inP?astheGrassmannian G=G(2,4)oflines inP®,wecanseeexplicitly what these twofamilies of 2-planes are:forevery point p€P®thelocus oflines passing through pforms‘22-planeonG,andforeveryplan:H<P?thelocusoflineslyinginHisa2plane inG.These arethetwofamilies; indeed, inthiscase wecangotwo steps further. First, weseefrom this that each ofthese families ispara- ‘metrized byP,sothat theconnected component PSO,€ oftheidentity inthegroup ofmotions ofP5preserving theGrassmannian actsonP?, giving ustheinverse inclusion PSO,C cPGL€. Second, under thePliicker embedding each ofthese families iscarried into acopy ofthequadratic Veronese embedding ofP®intoP?,giving ustheidentification ofthedirect sumfactors ofthe third exterior power ofthestandard representation ofs0g€ swith thesymmetric square ofthestandard representation ofs1,€. Exercise 19.1. Verify, without using theisomorphism with svgC and theanalysisabove,thatthestandardrepresentation Vofsi,Csatisfies ARV) =Sym?¥@Sym?V*. Notethatwehavenowidentified, inermsoftensorpowersofthe standard one, irreducible representations ofsogC with highest weight vectors Ly, L,+Ly+LyandLy+Ly~Lylying along theedge oftheWeyl chamber, aswell asonewith highest weight L+Lylying inaface. Wecanthus findirreducible representations withhighestweight7,ifnotforeveryyinAy0,atleast forevery weight 7intheintersection of#”with asublattice ofindex Din Ay, §19.2. Representations oftheEven Orthogonal Algebras Wewillnotexamine anyfurther representations ofsogC perse,leaving itasanexercisetodoso(andtocomparethe«sultsothecorresponding analysisforsi,€).Instead,wecannowdescribethegeneralpatternforrepresentations oftheeven orthogonal Liealgebras s0,,C. The complete story will have to wait until thefollowing lecture, since atpresent wecannot construct allthefepresentations of80,4(aswehavepointedout,wehavebeenabletodosointhecases n=2and 3studied sofaronly byvirtue ofisomorphisms with $19.2. Representations oftheEven Orthogonal Algebras 287 other Liealgebras; and there arenomore such isomorphisms from thispoint on).Wewillnonetheless give asmuch ofthepicture aswecan. Tobegin with, recall that theweight lattice ofs0,,C isgenerated by Ly,..., L,togetherwiththefurthervector(L,+--++L,)/2.TheWeylcham- ber, ontheother hand, isthecone WwW={Yalia, >a,2°>£a,}. Note that theWeyl chamber isasimplicial cone, with faces corresponding tothenplanes a,=a, ...,@,-, =a, and a,_, =~a,; theedges ofthe ‘Weyl chaniber urethus therays generated bythevectors Ly,Ly+La,-.-5 Lytort Lyn, Lytoe +L, and Ly++ +Lyy—L,(notethatLy+ +L,- isnotonanedge oftheWeyl chamber). Weseefrom thisthat, as inevery previous case, theintersection oftheweight lattice with theclosed Weyl cone isafreesemigroup generated byfundamental weights, inthiscase thevectors Ly,Ly+L2,.-., Ly+*"* +L,-2 and thevectors' aa(Ly te+L,2andB=(Ly+--+ Ly~L,V2 ‘Asbefore,theobviousplacetostarttolookforirreducible representations isamong theexterior powers ofthestandard representation. This almost works: wehave ‘Theorem 19.2. (i)Theexterior powers \*Vofthestandard representation Vof 80,,€ areirreducible fork=1,2,...,m~1;and(ii)Theexterior power N'V hasexactly twoirreducible factors. Proor, The proof will follow thesame lines asthat oftheanalogous theorem forthesymplectic Liealgebras inLecture 17;inparticular, wewill start by considering therestriction tothesame subalgebra asinthecase ofsp24C- Recall that thegroup Sp,,C ¢SL,,C ofautomorphisms preserving theskew form Qintroduced inLecture 16contains thesubgroup Gof antomorphisms ofthe space V=C2" preserving the decomposition V=Ce, .-.5€g}@Cfegsry+++»Can) acting asanarbitrary automorphism onthefirstfactor andastheinverse transpose ofthatautomorphism onthe second factor; inmatrices xX 0 = x - Infact, thesubgroup SO,,C cSL;,€ also contains thesame subgroup; we have, correspondingly asubalgebra "To conform tostandard conventions, with simple rots 6=fy~Lys for sm 1yand a=Ly+Lyt0haveo{ll,) =the fundamental weights, shout beputinte order: OnLybkbylorlsts'n~Qo OpAL ey LV aay ttLD 288 19,604C, £05C, andsox s-{6“Aeach60,6 isomorphic tosl,C. Denote byWthestandard representation ofs{,C. Asintheprevious case, therestriction ofthe standard representation Vof¢0,4€ tothesubalgebra s then splits v-wews intoadirect sumofWanditsdual; andwehave, correspondingly, Nv= @uewenwr ‘Wealso can sayhow each factor ontheright-hand side ofthis expression decomposes asarepresentation ofsl,C: wehave contraction maps Vy MW@NWA MW@NOWs, andthekernel of'¥,,,istheirreducible representation W with highest weight 2Ly +-*+ +2L,+Lyyy+°°+Ly.Therestriction ofA'Vto5is thus given by NV = wen,oh, where theactual highest weight factor inthesummand WM’ isthe vector WM eno NEA Cageggy AoA OgAOE NE FEA AA eanben RAagA(SUE Abya ‘Now,allthevectorsw'*-”havedistinctweights; anditfollows, asinExercise17.7, thatanyhighest weight vector fortheaction of803,€ onA'V will ea scalar multiple ofoneofthew'*". Itwillthus suffice, inorder toshow ihat /'V isirreducibleasrepresentation ofs0,,€fork<n,toexhibitforeach(a,b) with a+b <k<nother than (k,0) apositive root asuch that theimage g,(w*") ¥0.Thisissimplest inthecasea+b=k<(sothereisnofactor of@inwyjustasinthecaseofsp24€wehave Yorra-ver we) =(Euetan-nen —ExnenaveniMOr AAbeACaner AAFay) were 40 and¥,jisthegenerator ofthepositive rootspace 4,41, Incase a+b<k<n, weobserve first that forany iandj §19.2.Representations ofthe Even Orthogonal Algebras 289 YQ) =(Einrs —Ejns (CpArsp) arene #0 sothatwhenevera<i,j<n~b ¥,fw) SHAE A NOM eancnen AoA agAOME NA) FEAT AGA Canney AoA CaeAVisl(E(epAesg))*202) UkAaB) (eyAo NegAEAELACapers AACagAONE) #0. Itisalways possible tofind apair(i,j)satisfyingtheconditionsa<i,j<n—b since weareassuming a+b<k<n; thisconcludes theproofofpart(i). ‘The proof ofpart (ii)requires only onefurther step: wehave tocheck thevectorswis"witha+b=k=ntoseeifanyofthemmightbehighestweightvectorsfor502,€.Infact(asthestatement ofthetheoremimplies),twoofthem are:Itisnothard tocheck that, infact, w°' andw-!- arekilled byeverypositiverootspace4),.,,.Toseethatnoothervectorw®-""*js,lookattheactionOFYoya.ae2©Brynstan:WEhave Yorreoater) =(Exstunvata —EnvanvariMlr A°°ACaM Cavers A“ ACan) TELA AGG AEast ACnsaet AEnters AA Can TELA NA basa AOnsetAoACan #0. a Remarks, (i)This theorem will beaconsequence oftheWeyl character formula, which willtellusapriori that thedimension oftheirreducible representation ofo,4€withhighestweightL,+--++L,hasdimension(7) ifk<n,andhalfthatifk=n.(i)Note alsothatbytheabove, \"Visthe direct sumofthetwoirreducible representations T,,and Ty,with highest weights 22=Ly+--+ L,and B= Ly+0 +Ly—Ly.Indeed, theinclusion Ty,@Tay¢A'Vcanbe seen just from theweight diagram: /i”possessesahighestweightvectorwith highest weight L,+--* +L,,andsocontains acopy ofTq;butthisrepre- sentation does notpossess theweight 2f,andso\'Vmust contain Ty,as well. (Alternatively, weobserved inthepreceding lecture that inchoos- inganordering oftheroots wecould have chosen ourlinear functional |= cyHy+: +6,Hy with cy>cp>" >—c, >Owithout altering thepositive 290 19,sog€,$0,C,andso, rootsortheWeylchamber; inthiscasetheweightJof(*VwithI(2)maximalwould be28,showing thatTc AV) (ii) Ifwewant toavoid weight diagrams altogether, wecan still seethat NV must bereducible, because theaction ofs0,,C preserves two bilinear forms: first, wehave thebilinear form induced on/\*V bytheform QonV; and second wehave thewedge product ONY xNV+ AMV =C, thelastmaptakinge,°°»¢,4t0I.Itfollowsthat\*Visreducible;indeed, ifwewant toseethedirect sum decomposition asserted inthestatement of thetheorem wecanlook atthecomposition UAV NYE OMY, where thefirst map istheisomorphism given byQand thesecond is theisomorphism given byg.The square ofthismap istheidentity, and decomposing A'V into +1and —Ieigenspaces forthis map gives (wo subrepresentations. Exercise 19.3*, Part (i)ofTheorem 19.2canalso beproved byshowing that foranynonzero vector weMY, thelinear span ofthevectors X(w), for X€50,C, isallofA'V.Forthese purposes take, instead ofthebasis wehavebeenusing,anorthonormal basis0,..,0,forV=C",m=2n,s0Q(v,6)=44.Thevectors vy=0, °°"Aty,I={iy<-* <iy),form abasis forMY, and0,Chasabasis consisting ofendomorphisms V,.,p<q,which takes0,100,,0,(0~v4,andtakestheotherv,tozero.Compute theimagesV,«(0,),andprove theclaim, first, when w=e, forsome I,andthen byinduction on thenumber ofnonzero coefficients intheexpression w=¥a,0,. For(ii)a similar argument shows that \*Visanirreduciblerepresentationofthegroup O,C,andtheideasof§5.1(cf.§19.5)canbeusedtoseehowitdecomposes over thesubgroup SO,€ ofindex two. Wereturn now toouranalysis oftherepresentations ofs0,,C. Bythe theorem, theexterior powers ¥,A?¥, ...,/-?V provide uswith theirredue- iblerepresentations with highest weight thefundamental weight alongthefirst n~2 edges oftheWeyl chamber (ofcourse, theexterior power /'V isirreducible aswell,butaswehaveobserved, L,+°*:+L,..isnotonanedgeoftheWeyl chamber, and soA*“'V isnotasuseful forourpurposes). Forthe remaining twoedges, welave found irreducible representations with highest weights located there, nemely thetwo direct sum factors of/*V; but the highest weights ofthese tworepresentations arenotprimitive ones; they aredivisibleby2.Thus,giventhetheoremabove,weseethatwehaveconstructedexactly one-halftheirreduciblerepresentations of$0.4,n..mely,thosewhose highestweightliesinthesublatticeZ(I.,,...,Ly]©s\y.Explicitly,any weightyintheclosedWeylchambercanbeepressed(uniquely) intheform $19.2.Representations oftheEvenOrthogonal Algebras 291 yea hyto +ay(Ly +o +bya) dyg(LyHoeDyag~Dg2+ag(Ley+o+Ly2 with a,€N.Ifa,_, +a,iseven, with a,_, >a,weseethat therepresentation: Sym" V@---@Sym*-2(\"7V) @Sym*(A"""V) @Sym"? (Py9) willcontain anirreducible representation F,with highest weight y;whereas if 4,=4,4, wewillfindT,inside Sym"V@+@Sym*--(\-2V) @Sym"(A""'V) @Sym'o"*-7(P,), There remains theproblem ofconstructing irreducible representations T, whose highest weight 7involves anoddnumber ofasandf's.Todothis,we clearly have toexhibit irreducible representations I,and T,with highest weights «andB.These exist, andarecalled thespinrepresentations of602,C; wewillstudy them indetail inthefollowing lecture. Weseefrom theabove thatonce weexhibit thetwo representations T,and I,wewillhave con- structed alltherepresentations ofso,,C. Therepresentation I,with highest weight 7written above willbefound inthetensor product Sym"*V @---@Sym*-s(\"-7V)@Sym*-"(T)@Sym*(T). Forthetimebeing,wewillassumetheexistenceofthespinrepresentations ofs0,,C; there isagood deal wecansayabout these representations just on thebasis oftheir weight diagrams. Exercise 19.4*. Find theweights (with multiplicities) oftherepresentations AY,andalsoofPq,zp,Te,andFy, Exercise 19.5. Using theabove, show thatI,andTyaredual tooneanother when nisodd, and that they areself-dual when 1iseven. Exercise 19.6. Give thecomplete decomposition into irreducible representa~ tions ofSym?T, andA*T,, Show that LOT, =h, OM VON“*VON VO” Exercise 19.7. Show that TOT <M'VENVONVO Exercise 19.8, Verify directly theabove statements inthecase ofso,C, using. theisomorphism with s1,C. Exercise 19.9.Showthattheautomorphism ofC?*thatinterchanges e,andam, leaving theother e,fixed, determines anautomorphism ofso,,€ that preserves then—2 roots Ly—Ly,..y Ly-2—L,-1 and interchanges L,-1 —L,andL,-, +L,.Thisautomorphism takes therepresentation Vto. itself, butinterchanges I,andTy, 29 19,40,6,0,6,and46 §19.3. Representations ofs0,C While wemight reasonably beapprehensive about theprospect ofafamily of Liealgebras even more strangely behaved than theeven orthogonal algebras, thereissomegoodnews:eventhoughtherootssystemsoftheoddLiealgebras appear more complicated than those oftheeven, therepresentation theory of theoddalgebras issomewhat tamer. Wewilldescribe these representations, starting with theexample ofso,C; webegin, asalways, with apicture ofthe root diagram: r or ener ia Lith Sy ieel n> an | a Aswesaid, thislooks liketheroot diagram forspgC, except that theroots +2L,havebeenshortened to+L,.Unlikethecaseofs0,C,however, where thelong and short roots could beconfused and theroot diagram was corre- spondingly congruent tothat ofsp,C, inthepresent circumstance theroot diagram isnotsimilar toanyother; theLiealgebra s0,C, infact, isnot isomorphic toanyoftheothers wehave studied. Next, theWeyl chamber: neueky <4 --— oe hythy )4: | nn Again, theWeyl chamber itself looks just likethat ofspsC; thedifference inthispicture isintheweight lattice, which contains theadditional vector (Ly+Ly+Lyy/2. $193. Representations of#0, ca) Asusual, westart our study oftherepresentations ofso, with the standard sepresentation, whose weights are+L,and0: Ls <>ty ‘NotethatthehighestweightL,ofthisrepresentation liesalongthefrontedge oftheWeyl chamber. Next, theweights oftheexterior square A?V are £L;+Ly,+L, and0(taken three times); this, ofcourse, isjust theadjointrepresentation, NotethatthehighestweightL,+L,ofthis representation is the sameasthatofthe exterior squareofthestandardrepresentation fors0C, butbecause ofthesmaller Weyl chamber this weight does indeed lieonan ‘edge ofthechamber. Next, consider thethird exterior power A°V ofthestandard. This has weights +L, +L, +Ls,+L,+Lj,+L;(withmultiplicity 2)and0(with multiplicity 3),ie,atthemidpoints ofallthevertices,edges,andfacesofthe cube: - Lye tly lews te vet els. Lytls Ase tr8l-——5 -‘ a Itisnotobvious, from theweight diagram alone, that thisisanirreduciblerepresentation; itcouldbethat/\°Vcontainsacopyofthestandardrepresen-tationVandthattheirreducible representation F,,,+1,thushasmultiplicity {on theweights +L, and multiplicity 2(or1)at0.Wecan rule out this possibility bydirect calculation: forexample, ifthiswere thecase, then \?V would contain ahighest weight vector with weight L.The weight space with 19,s06.,€0,C,and60g cigenvalueL,inA°Visspannedbythetensors.e,Ae,Aesande,Aes0&6 however, and ifweapply tothese the generators Xy.1=Ey.2— Esa Xq,5=Es,3~Eo,sandUy=Es,~Er,90ftherootspacescorrespondingtothepositive roots Ly~La,Ly—Ly,andLs,weseethat Nally AesAee)=C1A€2ACis Usles Ae906)=erAesAe;#0: Xaaley Ae: Aes) =e, A-€2 Aees Usle, ney es)=0. There isthus nolinear combination ofe,ae)Aésand€,0€yAé,killed byboth U3andX;,3,showing thatA?hasnohighest weight vector ofweight Ly Exercise 19.10. Verify thatA?V does notcontain thetrivial representation. Wehave thus found irreducible representationsofso,Cwithhighestweight vectors along thethree edges oftheWeyl chamber, and asinthecase ofsogCwwehavetherebyestablished theexistenceoftheirreducible representations of 0,€ with highest weight inthesublattice (Ly,L2,Ly}.Tocompletethe description, weneed toknow that therepresentation T,with highest weight&=(Ly+Ly+Ly)/2exists, andwhatitlookslike,andthistimethereisno isomorphism toprovide this; wewillhave (owait until thefollowing lecture. Inthemeantime, wecanstillhave funplaying around both with therepresen- tations wedoknow exist, and also with those whose existence issimply asserted, Exercise19.11.Findthedecomposition intoirreducible representations ofthetensor product V@/\?V; inparticular findthemultiplicities oftheirreducible representation F3y,4,, Withhighest weight 21,+Ly. Exercise 19.12. Show that thesymmetric square oftherepresentation ¥ decomposes intocopy of A°Vandatrivial one-dimensional representation. Exercise 1913, Find thedecomposition into irreducible representations =f Mr. §19.4. Representations ofthe Odd Orthogonal Algebras Wewillnow describe asmuch aswecanofthegeneral pattern forrepresenta- tions oftheoddorthogonal Liealgebras £03,,,C. Asinthecase oftheevenorthogonal Liealgebras,theproofoftheexistencepartofthebasictheorem(14.18) (that is,theconstruction oftheitreducible representation with given $194, Representations oftheOdd Orthogonal Algebras 25 highest weight) willnotbecomplete until thefollowing lecture, butwecan work around thispretty well.Tobeginwith,recallthattheweightlatticeof50,4€is,likethatofs0,4C, generatedbyLy,...,L,togetherwithtefurthervector(Ly+++L,)/2.The Weyl chamber, ontheother hand, istinecone W=(Saba,202°20,20} ‘TheWeyl chamber isaswehave pointed outthesame asforsp2,C, thatis, itisasimplicial cone with faces corresponding tothenplanes a,=d3,..., 4,-, =a, and a, =0.The edges oftheWeyl chamber arethus therays generated bythevectors Ly,Ly+Lay.soyLy+77+byasandLy°°"+Ly (note that L,+--+ Ly. isonanedge oftheWeyl chamber). Again, theintersection oftheweightlatticewiththeclosedWeylconesafreesemigroup,inthis case generated bythefundamental weights w,=Ly,,=Ly+Ly, sess Op =LyHo Ly and the weight w,=a=(Ly+++L,)/2. Moreover, aswesaw inthecases ofso4C and s0,C, theexterior powers of thestandard representation doserve togenerate alltheirreducible representa- tions whose highest weights areinthesublattice Z(L,,...., L,}:in general we hhave thefollowing theorem. ‘Theorem 19.14. Fork=, ...,m,theexterior power AV ofthestandard represeniation Vof602944 isthe irreducible representation with highest weight Lyte by Proor. Wewillleave thisasanexercise; theproof isessentially thesame asin thecase of$04, with enough of1difference tomake itinteresting. =. We have thus constructed one-halfoftheirreduciblerepresentations of 029,,: anyweight yintheclosed Weyl chamber canbewritten yayLy+ay(Ly+La)etyg(Ly+o+bya)+gly+00+Ly)/2 with a,€N;and ifa, iseven, therepresentation Sym"V@++@Sym*(AMV)@Sym™?(\"V) willcontainanirreducible representation T,withhighestweighty.Wearestillmissing, however, anyrepresentation whose weights involve oddmultiples of 4;toconstruct these, weclearly hae toexhibit anirreducible representation T,with highest weight a.This exitts andiscalled (asinthecase oftheevenorthogonal Liealgebras)thespinrepresentation of$024.4C-Weseefromthe abovethat onceweexhibitthespinrepresentation F,wewillhaveconstructed alltherepresentations of$0244; forany7asabove thetensor Sym"V @+@Sym*=-1(A""'V) @Sym**(T,) willcontain acopy ofI, {Asinthecase ofthespin representation T,oftheeven orthogonal Lie algebras, wecansaysome things about Teven inadvance ofitsexplicit construction; forexample, wecandothefollowing exercises. 296 19,604C,£07C,and604 Exercise 19.15. Find theweights (with multiplicities) oftherepresentations AV, andalso ofTy Exercise 19.16. Give thecomplete decomposition into irreducible representa- tions ofSym? and/?T,. Show that LOT =NVON'VONVO ONVONY. Exercise 19.17. Verify directly theabove statements inthecase ofs05C, using theisomorphism with sp4€. §19.5. Weyl’s Construction forOrthogonal Groups ‘The same procedure wesaw inthesymplectic case canbeused toconstruct representations oftheorthogonal groups, thistime generalizing what wesaw directly for/'V in§§19.2 and 19.4, Forthesymmetric form QonV=C*,‘thesameformula(17.9)determines contractions fromV®toV®4—"),Denotetheintersection ofthekernels ofallthese contractions byV¥". Forany partition2=(A2++"2dy20)ofd,let SV =V8OSY. (19.18) Asbefore, thisisarepresentation oftheorthogonal group O,€ ofQ. ‘Theorem 19.19. Thespace Sy,,V isanirreducible representation ofOC; Stay¥ nonzero ifandonly ifthesumofthelengths ofthefirst twocolumns ofthe Young diagram of2isatmost m. ‘Thetensor power V%decomposes exactly asinLemma 17.15, withevery- thing thesame butreplacing thesymbol <d)by{1}.Inparticular, SV =Vey =Iimfey: Vs VI, Exercise 19.20. Verify that S,,V iszero when thesum ofthelengths ofthe firsttwocolumns isgreater than mbyshowing that M"V@AV@V4"#-™is contained inJ,¥,(V°%") when a+b>m.Show thatS,V isnotzero when thesum ofthelengths ofthefirst two colmns isatmost m. Exercise 19.21*. (i)Show that thekernel ofthecontraction from Sym*V to ‘Sym!"?1 istheirreducible representation Sy,V vfso4€ withhighest weight dLy (ii)Show that Sym'V=SV®SuaO°®Sy-ap)" where pisthelargest integer <d/2. $19. Wey"'s Construction forOrthogonal Groups 27 ‘The proof ofthetheorem proceeds exactly asin§17.3. The fundamentalfactfrominvarianttheoryisthesamestatement as(17.19),with,ofcourse,theoperators 3,=,©,defined usingthegivensymmetricform,andthegroup Spa.C replaced byO,C (and thesame reference toAppendix F.2forthe proof). The theorem then follows from Lemma 6.22 inexactly thesame way asforthesymplectic group. Tofindtheirreducible representations over SOC onecanproceed asin $5.1. Weyl calls twopartitions (each with thesum ofthefirsitwocolumn lengthsatmostm)associated ifthesumofthe lengths oftheir first columns is mand theother columns oftheir Young diagrams have thesame lengths. Representations ofassociated partitions restrict toisomorphic representa-tionsofSO,€.Notethatatleastoneofeachpairofassociated partitions willhave aYoung diagram with atmost 4mrows. Ifm=2n+1isodd, noAis associated toitself, butifm=2nis even, anyAwith aYoung diagram with n nonzero rows will beassociated toitself, and itsrestriction will bethesum of twoconjugate representations ofSO,,C ofthesame dimension. Thefinal result is ‘Theorem 19.22.(i)Ifm=2n+LandA=(2,>°°:&Ay2O)sthenSyyVisthe irreducible representation ofsoqC with highest weight 4yLy +°°"+Agloy (i)Ifm=2n,andA=(AyBe Ayay2OhthenSyyV istheirreducible representation ofs0,C with highest weight 4,Ly+" +Aghy(ii)Ifm=2n,and2=(Ay&*°2Ayg&Ay>0),themSyaVisthesumoftwoirreducible representations of$0,withhighestweightsAL,+°°"+AgLyandAyLy$00+AybytAnke Exercise 19.23. When misodd, show thatO,€ =SO,C x{£1}, Show thatitdandrareassociated, then«=A@¢,wherecisthesignofthedet:rminant. Wepostpone toLecture 25alldiscussion ofmultiplicities ofweigh spaces, ordecomposing tensor products orrestrictions tosubgroups.AswesawinLecture15forGL,CandinLecture17forSp.,C,itispossible tomake 2commutative algebra S!!= SMV) outofthesum ofollthe irreducible representations ofSO,C, where V=C"isthestandard repre- sentation. First supposem=2n+1isodd.Definethering'(V,n)asin§15.5, which isasum ofalltherepresentations S,(V) ofGL(V) where 2runs over allpartitions with atmost 1parts. Asinthesymplectic case, there isa canonical decomposition SAV) =SV ®Ja), andthedirect sumJ"=@,Jya(V) isanidealin$'(V, n).Thequotient ring SV)=AHM/S=@Sal) isacommutative graded ringwhich contains each irreducible representation 0fS0244,€ once. 298 19.s06C,£05C,ands0,€ Ifm= 2niseven, theabove quotient will contain each representation S,y(V/) twice ifhasnrows. Tocutitdown sothere isonly oneofeach, one ‘canaddtoJ"relations oftheform x—1(x), forx€A'V, where 1:'V+AV istheisomorphism described intheremark (ii)afier theproof ofTheorem19.2,Foradetaileddiscussion, withexplicitgenerators fortheideas,see[L-T]. LEC URE 20 Spin Representations ofso,,C Inthislecture wecomplete thepicture oftherep:esentations oftheorthogonal Lie algebras byconstructing thespin representations S*ofs0,€; this also yields a description ofthespin groups SpingC. Since therepresentation-theoretic analysis ofthespacesS*wascarriedoutinthepreceding lecture,weareconcernedhereprimarily withthealgebrainvolvedintheirconstruction. Thus,§20.1and§20.2,whileelementary, involve some fairly serious algebra, Section 203, where webriefly sketch thenotion of trait, may seem mysterious tothereader (this iatleast inpart ecause itissototheauthors)iso,tmaybeskipped.Finally,weshouldsaythatthesubjectofthespin representations of4,€ isavery rich One, and one that accommodates many diferent Points ofview thereader who isintrested isencouraged t0trysome oftheotherApproacts thatmaybefoundintheliterature. §2011: Clilford algebras andspin representations of0,C §202: Thespin groups SpingC andSpin, $203: SpingC and triality §20.1. Clifford Algebras and Spin Representations ofs0,€ Webegin thissection bytrying tomotivate thedefinition ofClifford algebras. ‘Wemay begin byasking, why were weable tofindalltherepresentations of SL,CotSp2,Cinsidetensorpowersofthestandardrepresentation, butonly halftherepresentations ofSO,,€ arise thisway? One difference thatpoints inthisdirection liesinthetopology ofthesegroups:SL,CandSp2,€aresimplyconnected, while SO, hasfundamental group Z/2form> 2(forproof: see§23.1). Therefore SO,C hasadouble covering, thespin group Spin,.C. (For m<6,these covering’ could also beextracted from ouridentifications aw 20,SpinRepresentations ofso oftheadjointgroupPSO,Cwiththeadjointgroupofothersimplyconnectedgroups; eg.thedouble cover ofSOC isSLC.) Wewillseethatthemissing representations arethose representations ofSpin, that donotcome from representations ofSO,,C. This double covering may bemost readily visible; andprobably familiar,forthecaseofthe realsubgroup SO,Rofrotations: arotation isspecified by anaxis torotate about, given byaunit vector u,and anangle ofrotation about u;thetwochoices -twofunit vector give atwo-sheeted covering. In other words, ifDis theunitballinR°,there isadouble covering S*=D'/aD? +50, which sends avector »inD?torotation bytheangle 2r[nl] about theunit vector v/lol (theorigin andtheunitsphere OD!aresent totheidentity transformation). This covering iseven easier toseefortheentire orthogonal group O,R, which isgenerated byreflections R,inunit vectors v(with tvdetermining thesame reflection): wecandescribe thedouble cover ofO,R asthegroup generated byunit vectors v,with relations Op ety Na My whenever thecompositions ofthecorresponding reflections areequal, ie, whenever Ry,07770Ry,=Ry,07Ry, and also relations (-0:(=m) =0-w forallpairs ofunit vectors vand w.(Note that ifwerestricted ourselves to products ofeven numbers ofthegenerators v€0D?wewould getback the double cover ofthespecial orthogonal group SO,C) Howshouldwegeneralizethis?Theanswerisnotobsiou:.Foronething, {forvarious reasons wewill not trytoconstruct directly agroup that covers theorthogonal group ingeneral. Instead, given avector space V(real or complex) and aquadratic form Qon¥,wewill first construct analgebra Cli, Q),called theCliford algebra. The algebra Cliff(¥, Q)willthen turn 201. Clifford Algebras andSpin Representations of04 301 outtocontain initsmultiplicative group asubgroup which isadouble coveroftheorthogonal groupO(V,Q)ofautomorphisms ofVpreserving Q.Byanalogywiththeconstruction ofthedoublecoverofSOR,theClifford algebra Cliff(V, Q)associated tothepair (V,Q) isanassociative algebra containing and generated byV.(When wewant todescribe thespin group inside CliM(V, Q)wewill restrict ourse'ves toproducts ofeven numbers of elementsofVhavingafixednormQ(®,»)ifoddproductsareallowedaswell,Wwegetagroupcalled“Pin”whichisadoublecoveringofthewholeorthogonalgroup.) Tomotivate thedefinition, wewould likeCliff(V, Q)tobethealgebra generated byVsubject torelations analogous tothose above forthedouble cover oftheorthogonal group. Inparticular, foranyvector vwith Q(e, v)=1, since thereflection R,inthehyperplane perpendicular tovisaninvolution, we want vont inClifV, Q).Bypolarization, thisisthesame asimposing therelation vow +we =20(0,») {orall vand winV.Inparticular, w-v =~v-w ifand wareperpendicular. Infact, theClifford algebra' willbedefinedbelowtobetheassociativealgebra generated byVand subject totheequation v-v=Q(o, »).Lookingahead,wewillseelaterinthissectionthateachcomplexCliffordalgebracontainsanorthogonal Liealgebraasasubalgebra, Thekeytheorem.isthenthatClif(V,Q)isisomorphic eithertoamatrixalgebraorto.asumoftwomatrix algebras. This inturn determines either oneortwo representations of theorthogonal Liealgebras, which turn outtobetherepresentations whichwereneededtocompletethestoryinthelastlecture,Justasinthespeciallinear andsymplectic cases, thecorresponding Liegroups arenotreally needed to construct therepresentations; they canbewritten down directly from theLie algebra. Inthis section wedothis, using theClifford algebras toconstructtheserepresentations ofs0,€directly,andverifythattheygivethemissingspin representations. Inthesecond section ofthis lecture wewillshow how thespin groups sitassubgroups intheir multiplicative groups. Clifford Algebras Given asymmetric bilinear form Qonavector space V,theCliford algebra C= C(Q) =Ciif(¥, Q)isanassociative algebra with unit 1,which contains andisgenerated byV,with v-v=Q(e, 0):Iforall €V.Equivalently, wehave theequation v-w-+wv=20(0,wh, (20.1) "Themathematical worldseemstobeaboutevenlydividedaboutthechoiceofsignshere,and‘onemusttranslatefromQt0=Qtogofromonesidetotheother. sue 20,SpinRepresentations ofso, forallvand winV.The Clifford algebra canbedeiined tobetheuniversal algebra with this property: ifEisany associative algebra with unit, and alinearmappingj:V-+Eisgivensuchthatj(0)*=Q(v,»)|forallveV,orequivalently Jt0)-l09) +j6)-s(0) =20(0, w)-1 (202) forallo, w€V,then there should bea unique homomorphism ofalgebras from C(Q) toEextending j.The Clifford algebra canbeconstructed quickly by taking thetensor algebra TI) = QV" =COVOVENOVOVOV)O-, andsetting C(Q) =T'(V)/1(Q), where 1(Q) isthetwo-sided ideal generated by allelements oftheform »@v—Q(v,0)°1. Itisautomatic that this CQ) satisfies therequired universal property. The facts that thedimension ofCis2,where m= dim(V), and that the canonical mapping from VtoCisanembedding, arepart ofthefollowing Jemma: Lemma 20.3. If€,...,€form abasis forV,then theproducts e,= 644"y'eis forT={iySig<0 <i), andwitheg=1,formahasisfor C(Q) =Cliff, Q). Proor. From theequations e,-¢, +"e;=20(¢, ¢)itfollows immediately that theelements e,generate C(Q). Their independence isnothard toverily directly; italso follows byseeing that theimages inthematrix algebras under themappings constructed below areindependent. For another proof, note that when Q=0,theClifford algebra isjust theexterior algebra 'V. In ‘general, theClifford algebra canbefilteredbysubspacesF;,consistingofthose elements which can bewritten assums ofatmost kproducts ofelements in V;onechecks that theassociated graded space Fi/F,y, isAMV. Forathird proof, one can verify that theClifford algebra ofthedirect sum oftwo orthogonal spaces istheskew commutative tensor product oftheClifford algebras ofthetwo spaces (cf.Exercise B.9), which reduces one tothetrivialcasewheredimV=1 a Since theideal 1(Q)<T(V)isgenerated byelementsofeven degree, the Clifford algebra inherits aZ/22 grading: C=C OCH=C OC, with CF CtEC*, C7 CO“, CCF >,CCECF;C*isspanned byproductsofanevennumberofelementsinVandC~isspanned byjroducts ofanoddnumber. Inparticular, C" isasubalgebra ofdimension 2"SinceC(Q)isanassociative algebra,itdetermines aLiealgebra,withbracket (a,b]=a:b—b-a.Fromnow onweassume Qisnondegenerate, The new representations ofs0,,C willbefound intwo steps §20.1. Cliford Algebras and Spin Reprosentations of60, 303 (embeddingtheLiealgebras0((:=0,insidetheLiealgebraoftheeven partofthe Clifford algebra CG;(i)identifying theCliffordalgebraswithoneortwocopiesofmatrixalgebras. Tocarry outthefirststep wemake explicit theisomorphism of\?¥ with 0(Q) that wehave discussed before, Recall that 80(Q) ={¥€End(V): (Xv, w)+Q(v,Xw)=0forallv,winV}. ‘The isomorphism isgiven by MVS s0(Q)< End(V), aAE+Pare foraand binV,where aay isdefined by Parv(0) =2(O(b, v)a—Qa, »)b). (20.4) Itisasimple verification thatgis ins0(Q). One seesthat thenatural bases correspond uptoscalars, eg,€Aey.Maps to2(E,,)~ Eysjasth $0themap isanisomorphism. (The choice ofscalar factor isunimportant here; itwas ‘chosen tosimplify later formulas) One calculates what thebracket on/?V must betomake thisanisomorphism ofLiealgebras: [ents Penal) =Pend? Penal?) —Pena? Pans) =anolQld,vJe—O(c,0)d)—2eenalQlb, va—Ola,v)b) =40(4, 11016, c)a— Ola, 0) =40(c, (Q(b, d)a—Ola, d)b) —4Q(b, v)(QUd, ade~Ofc, ahd) +4Q(a, w)(Q(d, be—Qc, Bd) =20(b, Penal) ~20(b, d)Penc(0) =29(a, d)g.r4(0) +20(a, e)Pens(0). This gives anexplicit formula forthebracket onA?V: [ar bend] =20(b, can d—20(b, dane ~20(a, den b+20a, edab. (20.5) ‘Ontheother hand, thebracket intheClifford algebra satisfies la-b.e-d] = ached —e-d-a-h =(2Q(b, chard~ave-b-d) ~(2Q(a,d)e-b~c-adb) =20(b, e)a-d —(20(b, d)a-e ~a-e-d-b) —29a, d)e-b +(20(a, c)-d-b —a-e-d-b) =20(b, e)a-d —29(b, d)a-e —29(a, deb +20(a,c)-ab, 304 20.SpinRepresentations of60,€ Itfollows thatthemap y:\?V+Cliff(V,Q)definedby Ua.»b)=Ha-b—b-a)=a-b~O(a,b) (96) isamap? ofLiealgebras, andbylooking atbasis elements again oneseesthat itisanembedding, This proves: Lemma 20.7. Themapping y0p"*: s0(Q)+C(Q)"**embedss0(Q)asaLie subalgebra ofC(Q)"**. Exercise208,Showthattheimageofyis Fyn Cy oKer(trace), where F,isthesubspace ofC(Q) spanned byproducts ofatmost twoelements ofV,andthetrace ofanelement ofC(Q) isthetrace ofleftmultiplication by thatelement onC(Q). Weconsider first theeven case: write V= W@W’, where Wand W’are n-dimensional isotropic spaces forQ.(Recall thataspace isisotropic when Q restricts tothezeroform onit)With ourchoice ofstandard QonV=C*, Wan betaken tobethespace spanned bythefirstnbasis vectors, W’bythe fast Lemma 20.9. Thedecomposition V=W@)W’determines anisomorphism of algebras C(Q) =End('W), whereNW=PW @NW. Proor. Mapping C(Q) tothealgebra E=Fnd(/'W¥) isthesame asdefining 1linear mapping from VtoE,satisfying (20.2). Wemust construct maps I:W-+ Band: W' +Esuch that KwF =O, Mw'P=0, (20.10) and Hos) 0Hw’) +10") 0Iw)=200», wT foranyweW,w’eW’,ForeachweW,letL,,€Ebeleftmultiplication by wontheexterioralgebraAW: LAG=wad, FeNW. For 9W®, letDy€Ebethederivation ofAWsuch that Ds(1) =0,Ds(w) =Sw)eA°W=CforweW=A'W,and 2Notethatthebilinearformygivenby(206)isalternatingsince(aa)=0,50itdefinesalinear map onAY, $201,CliffordAlgebrasandSpinRepresentations ofs0,€ 305 DAE A8)=DoE) 0E+(=DHE ADylC). Explicitly, Dy(wy 0°0.0) =S(—19m)(oy,AAW,AAw).Now set Iw) =Ly, Mw!) =Ds, (20.11) where 9©W*isdefined bytheidentity 9(w) =20(w, w’)forallweW.The re- quired equations (20.10) arestraightforward verifications: onechecks directly onelements inW=A'W, andthen that, iftheyhold on{andé,they hold on A&.Finally, onemay seethat theresulting map isanisomorphism by looking atwhat happens toabasis. Oo Exercise 20.12.TheleftC(Q}-module A'Wisisomorphic toaleftidealinC(Q). Show that iffisagenerator forA"W’, then C(Q)-f =\'W-f, andthemap (++E-f gives anisomorphism KWNWf =CO) ofleftC(Q}modules. Now wehave adecomposition AW =A\"**W @AW into thesum of even andoddexterior powers, andC(W)"** respects thissplitting. Wededuce from Lemma 20.9 anisomorphism Cyr" =End(A"W) ®End(A“W). (20.13) Combining with Lemma 20.7, wenow have anembedding ofLiealgebras: 80(Q) <C(Q)™*" &all") @gl), (20.14) andhencewehavetworepresentations ofs0(Q)=s0,,C,whichwedenoteby St=Avw and Sy=AmW. Proposition 20.15. Therepresentations S*aretheirreducible representationsof '80,,Cwithhighestweightsa=\(Ly+°*+L,)andB=\(Ly+++"Lys= L,).More precisely, St=0, and S=Ty ifniseven; St=, and S-=0, sfnisodd. Proor. Weshow thatthenatural basis vectors e,=¢),A-""Ae, forNW areweight vectors. Tracing through theisomorphisms established above, we seethatHy=E,,— Exyigssin b€$0,,€ corresponds to4(e,Aéys,)inMV, which corresponds toI(¢,"e,,4~ 1)inC(Q), which maps to ML, Daey~1)=Le,©Dez~4€End(NW). ‘Asimple calculation shows that 306 20,SpinRepresentations of60,€ eyifter baeDaten)=iftig. Therefore, ¢,spansaweightspacewithweight(Seri~S414)Allsuch weights with given |1|mod 2arecongruent bytheWeyl group, soeach of St=Av" andS=A“4W must beanirreducible representation. The highest weights areeasy toread off.Forexample, thehighest weight forN**Wis3.L,=aifmiseven,whileifmisodd,itshighestweightisp. These two representations S*and S~areusually called thehalf-spin representations of$03,C, while their sum $=S*@S~ =A'W iscalled thespinrepresentation. Frequently, especiallywhenwespeakoftheevenandoddccases together, wecallthem allsimply “spin representations.” Elements ofSarecalledspinors.Forotherproofsoftheproposition seeExercises20.34and2035. For the odd case, write V=W®W'®U, where Wand W" are mn dimensional isotropic subspaces, andUisaone-dimensional space perpendic- ulartothem. Forourstandard Qon€?**!, these arespanned bythefistn, thesecond n,and thelast basis vector. Lemma 20.16. The decomposition V=W@W’@Udeterminesanisomor- phism ofalgebras C(Q) =End NW)@End(AW). Proor. Proceedingasintheevencase,tomapVtoE=End(\'W),map weW toL,,w’€ W"toDs,where 9{w) ==20(w, w’)asbefore, Letugbetheelement inU'such that O(u, uo)=1,andsend tytotheendomorphism that isthe identity onA***W, andminus theidentity onA%*W. Since thisinvolution skew commutes with allLandD,,theresulting map from V=W@ W’@U toEdetermines analgebra homomorphism from C(Q) toE.The map to End(\'W’) isdefined similarly, reversing theroles ofIVand W’.Again one checks that themap isanisomorphism bylooking atbases. a Exercise 20.17*. Find agenerator foraleftideal ofC(Q) that isisomorphic toAW. Thesubalgebra C(Q)"™* ofC(Q) ismapped isomorphically onto either of thefactors bytheisomorphism ofthelemma, sowehave anisomorphism in the odd case C(gy""* =End(A'W). (20.18) AAsbefore, thisgives arepresentation S=:'W ofLiealgebras: $0344, =80(Q) &C(Q*** =gl W)=gl(S). (20.19) $20.2, tne Spin Groups Spingt: and Spin, 307 Proposition 20.20. Therepresentation S=/'W isthetrreductble representationOf$0294,withhighestweight aaKL, + +L). Proor. Exactly asintheeven case, each ¢,isaneigenvector with with weight HeerLi~yarLy)ThistimeallsuchweightsarecongruentbytheWeyl ‘group,sothismustbeanirreducible representation, andthehighestweight isclearly4(L,+--+L,). oO ‘Aswesaw inLecture 19,theconstruction ofthis spin representation $ finishes theproof oftheexistence theorem forrepresentations ofso,C, and hhence foralloftheclassical complex semisimple Liealgebras. Exercise 20.21%. Use theabove idcntification oftheClifford algebras with ‘matrix algebras(ordirectcalculation) tocomputetheircenters.Inparticular, show that theintersection ofthecexter ofCwith theeven subalgebra Cis,alwaystheone-dimensional space©»scalars.ShowsimilarlythatifxisinC™!#andx-v=—o-xforallvin¥,thenx=0. Exercise 20.22*. ForX€s0(Q) andveV,wehave X-ve Vbythestandard action ofs0(Q) onV.Ontheother hand, w>have identified s0(Q) and Vas subspaces oftheClifford algebra C,sowecancompute thecommutator [X,0]. Show that these agree: Xev=[XeJeVeG. Problem 20.23*. LetC(p, q)betherealClifford algebra corresponding tothe quadratic form with ppositive and qnegative eigenvalues, Lemmas 20.9 and 20.16 actually construct isomorphisms ofC(n, n)with arealmatrix algebra,andofC(n+1,n)withaproductoftworealmatrixalgebras.ComputeC(p,4) forother pand q.Allareproducts ofoneortwo matrix algebras over R,C, or. §20.2. The Spin Groups Spin,,C and Spin, ‘The Clifford algebra C=C(Q) isgenerated bythesubspace V=C%,and C hasananti-involution xt x*,determined by (pO =(=op forany v,,..., »,inV.This operation »,sometimes called theconjugation, is thecomposite of: themain antiautomorphism orreversing map t:C+ Cdetermined by HO, oO)=DE (20.24) forby, .0.5,inVand themain involution awhich istheidentity onC**** andminus theidentity onCie, Cted (20.25) Notethatx:y)*=y*-x*,whichcomesfromtheidentitiest(x-y)=¢(y)-1(x) and a(x:y)=a(x)-a(y). Exercise 20.26. Use theuniversal property forCtoverify that these arewelldefined:showthataisahomomorphism fromCtoCandtisawell-definedhomomorphism from Ctotheopposite algebra ofC(thealgebra with the ‘same vector space structure, butwith reversed multiplication: xy=y-x) Insteadofdefiningthespingroupasthesetofproductsofcertain elements of¥,itwill beconvenient tostart with amore abstract definition. Set Spin)={xeC(O)":x-x*=Landx-Vext eV}.(2027) Weseefromthisdefinition thatSpin(@)formsaclosedsubgroupofthegroupofunits inthe(even) Clifford algebra. AnyxinSpin(Q) determines anendo- morphism p(x) ofVby px}(o)=x-0-x*, vev, Proposition 20.28. Forx€Spin(Q),p(x)isinSO(Q).Themapping p:Spin(Q) -SO(Q) isahomomorphism, making Spin(Q).a connected two-sheeted covering ofSO(Q). Thekernelofpis{1,—1}. Proor.Wewillprovesomething more.Definealargersubgroup, thistimeofthemultiplicative group ofC(Q), by Pin(Q) ={xe C(Q):x-x* =Land x-Vex*V}, (20.29) anddefine ahomomorphism P:Pin(Q)— OQ), ()(0) =a(x)-0-x*, (20.30) where a:C(Q)-»C(Q)isthemaininvetion. Toseethat p(x) preserves thequad. uticform Q,weusethefactthat forw inV,Q(w, w)=w-w =—w-w!, and calculate: ApxI(0),p(x)(0))=—a(x)-v-x* (a(a)-v-x4) =a(x) v-xtxt a(x)? ==a(x)-v-0* ae) =QC, v)atx)-a(x*) =Qo, wlalx-x*) =Ql0, Weclaim next that pissurjective. This follows from thestandard fact(see Exercise 20.32) that theorthogonal group O(Q) isgenerated byreflections. Indeed, ifR,,isthereflection inthehyperplane perpendicular toavector w, normalized $0that Q(w, w)=—1, itiseasy toseethat wisinPin(Q) and Ow) =Rysinfact, wewe =we(—w) =—Qlw, w)=1, and so 1)(8)=a(w):wow"=—w-l=ws andifQ(w, v)=0, OW)(0)=a(w)-0-wt=—w-v-we =v-wewe=o. ‘The next claim isthat thekernel ofpcuthelarger group Pin(Q) is+1.Suppose xisinthekernel,andwritex=xo+x,Withx9€Candx,€C4 Thenx9-0=0:x9forallv€¥,$0x9isinthecenterofC.Andx,-v=—0-x, forallv¢V.ByExercise20.21,xqisinC-1,andx,=0.S0.x=xisinCand x= ljsox= +1 ItfollowsthatifReO(Q)iswrittenasaproductofreflections R,.,°.-.°Re, then thetwo elementsinp-"(R)are+,...,Inparticular,wegetanother description ofthespin groups: Spin(Q) =Pin(Q)>C(Q)"*"=p~(SO(Q)) =Cbwewn EKOO) ==. 2031) Since ~1=v-v forany vwith Qo, »)=—1, wesee that the spin group consists ofeven products ofsuch elements. ‘Tocomplete theproof,wemustcheckthatSpin(Q)isconnectedor,equiva- lently, that thetwo elements inthekernel ofpcan beconnected byapath. Weleave thisnow asanexercise, since much more willbeseen shortly.) Exercise 20.32. LetQbeanondegenerate symmetric bilinear form onareal orcomplex vector space V. (a)Show that ifvand warevectors inVwith Q(v, v)=Q(w, w)#0, thenthereiseitherareflection oraproductoftwo reflections that takes vinto w. (b)DeducethateveryelementoftheorthogonalgroupofQcanbewritten astheproduct ofatmost 2dim(V) reflections. Exercise 20.33*. Since Spin(Q) isasubgroup ofthemultiplicative group of C(Q),itsLiealgebraisasubalgebra ofC(Q)withitsusualbracket,Verifythatthissubalgebra isthesubalgebra so(Q) that wasconstructed in§20.1. Exercise 20.34. The factthat /'W (and \'W" intheoddcase) isanirreducible module over C(Q) isequivalent tothefactthatitisanirreducible representationofthe group Pin(@Q) since thelinear span ofPin(Q) isdens= inC(Q). (a)Apply theanalysis of§5.1tothesubgroup Spin(Q) <Pin(Q) ofindex two. Intheodd case, NW and ‘VW’ areconjugate representations, sotheir restrictions toSpin(Q) areisomorphic andirreducible: thisisthespin representation. Intheeven case, /'W isself-conjugate, and itsrestriction to Spin(Q) isasum oftwoconjugate irreducible representations, which arethe twohalf-spin representations. (b)OFtherepresentations ofSpin(Q) (ie., therepresentations ofso,C), which induce irreducible representations ofPin(Q) andwhich arerestrictionsofirreducible representations ofPin(Q)? Exercise 20.35. Deduce theirreducibility ofthespin andhalf-spin represen- {ations from thefactthat their restrictions tothe2-groups ofExercise 3.9are irreducible representations ofthese finite groups. Exercise20.36%.ShowthatthecenterofSpin,(C)isp-"(I)={41}ifmisodd.If'm iseven show that the center is et =(£1, to}, where, interms ofour standard basis, ten ener—Hennes, MnCan—Hany aa 2 Exercise 20.37*. Show that thespin representation Spin(Q) +GL(S) maps into thespecial linear group SL(S). Show that form= 2nandneven, the half-spin representations also map into thespecial linear groups SL(S*) and SLs). Exercise 20,38*. Construct anondegenerate bilinear pairing fon thespinor space S=NW bychoosing anisomorphism ofA*Wwith Candletting As,)betheimageofr(3)a1€AWbytheprojection to\'W=C,wherexisthemain antiautomorphism). {a)When m=2n,show thatcan alsobedefined bytheidentity is,Of= u(s:f):t-f foranappropriate generator fofAW’. Deduce that theaction ofSpin(Q) onSrespects thebilinear form f. (b)Show that fis symmetric ifniscongruent to0or3modulo 4,and skew-symmetric otherwise. Sothespin representation isahomomorphism Spiny... SOC ifn=0,3(4), Spinz,.,C +SpyC ifn=1,2(4). $202. TheSpin Groups Sing andSpin, al (©)Ifm=2n,therestrictions offtoS*and S~arezero ifnisodd. For n even, deduce thatthe half-spin representations arehomomorphisms Spiny,C-» 80; 1C ifn=0(4), Spin,,€ +SpriC ifm=2(4). Note inparticular that SpingC hastwo maps toSOsC inaddition tothe original covering. “Triality,” which wediscuss inthenext section, describes therelation among these three homomorphisms. Exercise 20.39. Show that thespin and half-spin representations give the isomorphisms wehave seen before: Spin,C =GL(S*) =GL,C =C*, Spin ~SL(S) =SLC, SpingC =SL(S*) xSL(S)=SL3CxSL3C, Spin, =SpIS) =Sp.C, Spinge &SL{S*) =SLC. Exercise 20.40. LetC,,denote theClifford algebra ofthevector space C™with ourstandard quadratic form Ox (a)ThembeddingofC7"=W®W'inC?**!=W®W’@Uasindicated induces inembedding ofCz,inCyay1, andcorresponding embedding of Spin,,€ inSpin,.,C andofSO,,€ inSOy_4,C. Show thatthespinrepresen- tation$ofSping,,y€restricttothespinrepresentation $*@S~ofping.C. (b)Similarly there isanembedding ofSpin,,,,C inSpin,,,2C comingfromanembedding ofC™"!=W@W"@Vine"?=W@WOU;@UsshereU,@U;=C@C withthequadraticform(oandU=Cis ‘roi) embedded inU,®U,bysending |to ry J.ShowthateachoftheER half-spin representations ofSpin,,,.° restricts tothespin representation of Spin3,4,€. Verylittleoftheabovediscussion needstobechangedtoconstruct therealspin groups Spin,,(R), which aredouble coverings oftherealorthogonal groups SO,,(R). One uses therealClifford algebra Cliff(R™, Q)associated to therealquadratic formQ=—Q,., where Q,.is thestandard positive definite quadratic form onR™.Ifv,areanorthonormal basis, theproducts inthis Clifford algebra aregiven by ney =orn, HAR and opy=—b 32 20,SpinRepresentations of90,€ ‘The same definitions can begiven asinthecomplex case, giving riseto coverings Pin,(R) ofO,(R) andSpin,(R) ofSO,(R). Exercise 2041, Show thatSpin,,R isconnected byshowing thatifvandware any twoperpendicular elements inVwith Q(o, x)=Q(w, w)=—1,thepath 1Hs(cos(tpn +sin(Qw)-(cos(t}v ~sin(dw), OSt<H/2 connects —1 to1 Exercise20.42.Showthati+,"0,jh0y"0,,kh+040,determines aniso-morphism ofthequaternions Ifonto theeven partofCliff(R’, —Q,), such that conjugation ~inHcorresponds totheconjugation intheClifford algebra. Show that thismaps Sp(2) ={q€Hl4j =1}isomorphically onto Spiny, and that this isomorphism iscompatible with themap toSO,R defined inExercise 7.15. MoregenerallyifQsaquadraticformonR™withppositiveandqnegativecigenvalues, wegetagroup Spin*(p, q)intheClifford algebra C(p, 4)= Ciff(2", 0),with double coverings Spin"(p, q)+$0"(p, 9) Exercise 20.43*, Show that Spin*(p,q) isconnected ifpandqarepositive, except forthecase p=q=1,when ithastwo components. Show that ifin thedefinition ofspin groups one relaxes thecondition x:x* =1tothe condition x-x* =+1,onegets coverings Spin(p, 4)ofSO(p, 4. §20.3. SpingC and Triality When miseven, there isalways anouter automorphism ofSpin,(C) that interchanges thetwospin representations S*andS~,while preserving the basic representation V=©"(efExercise 19.9). Incasem=8,allthreeofthese representations V,S*,audS~areeight dimensional. One basic expression of, trialty isthefact that there areautomorphisms ofSpingC ofso, that permute these three representations arbitrarily. (Infact, thegroup ofouter automorphisms modulo inner automorphisms isthesymmetric group onthreeelements.) Wegiveabriefdiscussion ofthisphenomenon inthissection,inthe form ofanextended exercise. Toseewhere these automorphisms might come from, consider thefour simple roots: a= Ly~Ly,a= by—LyayeLly—Ly, ayLy+Ly. Note that a},a3,anda,aremutually perpendicular, and that each makes an angle of120° with a, $203.Sping€andTriality 33 Exercise 20.44*.Foreachofthesixpermutations of{a,,a5,a4}findtheorthogonal automorphism oftheroot space which fixes «,andrealizes the permutation ofa,a3,and a4. Each automorphism ofthisexercise corresponds toanautomorphism of theCartan subalgebra b.Inthenext lecture wewill seethat such auto- morphisms can beextended (nonuniquely) toautomorphisms ofthe Lie algebra so,(C). (For explicit formulas sec[Ca2],) ‘There isalso apurely geometric notion oftriality, Recall that aneven- dimensional quadric Qcancontain linear spaces Aofatmost halfthedimen- sion ofQ,and that there aretwo families oflinear spaces ofthis maximal dimension (cf.[G-H], [Ha]). Incase Qissix-dimensional, each ofthese families canthemselves berealized assix-dimensional quadrics, whichwemay. denote byQ*andQ™(seebelow). Moreover, there arecorrespondences that assign toapoint ofanyoneofthese quadrics a3-plane ineach oftheothers: Point in@ ——+ 3-plane inQ* be PeaPoint inQ*——+ 3-plane inQ Given P<Q,{A€Q*:Acontains P}isa3-planeinQ*,and{AeQ™:A contains P}‘sa3-plane inQ™. GivenA Q*,Aitselfisa3-planeinQ,and{T¢Q™:PAisa2-plane} isa3-plane inQ”. GivenA€Q~,Aitselfisa3-planeinQ,and{T¢Q*:PoAisa2-plane} isa3-plane inQ*. Torelate these twonotions oftriality, take Qtobeourstandard quadric inP’=P(V), with V=W@W’ with ourusual quadratic space, and let S*=Aw"WandS”=A°“Wbethetwospinrepresentations. InExercise 20.38 weconstructed quadratic forms onS*andS™,bychoosing aniso- morphism ofA*W with C.This gives ustwoquadrics Q*andQ~inP(S*) ne ‘Toidentify Q*andQ~with thefamilies of3-planes inQ,recall theaction ofVonS=\'W=S* @S~ which gave risetotheisomorphism ofthe Clifford algebra with End(S) (cf.Lemma 20.9). This infact maps S*toS~ 34 20.Spin Representations of#0, andS~toS*;so wehave bilinear maps VxSt9S"andVxS~St (20.46) Exercise20.47,ShowthatforeachpointinQ",represented byavectors€S*,{veV:-s=0}isanisotropic 4-planeinV,andhencedetermines aprojective3:planeinQ.Similarly, eachpointinQ~determines a3-planeinQ.Showthatevery 3-plane inQarises uniquely inoneofthese ways. Let ¢,>ydenote thesymmetric form corresponding tothequadratic form in¥,andsimilarly for$*and$~.Define aproduct StKS VYsxtrst, 20.48) byrequiring that <0, 5-0)=(0-5, Qs:forallve¥. Exercise 20.49. Usethisproduct, together with those in(20.46), toshow that theother four arrows inthehexagon (20.45) forgeometric triality canbe described asinthepreceding exercise. This leads toanalgebraic version oftriality, which wesketch following, [Ch2}. The above products determine acommutative but nonassociative product onthedirect sumA=V@S*®S~.Theoperation (0,5, 04 40'S,Dy determines acubic form onA,which bypolarization determines asymmetric trilinear form ®onA. Exercise 20.50*. One canconstruct anautomorphism JofAoforder three thatsends VtoS*,S*toS~,andS~toV,preserving theit quadratic forms, and compatible with thecubic form. The definition ofJdepends onthe choiceofanelementv,€Vands,€S*with<0),0,dy=(51.81)s+=1ssetty=vy-Siy80that(ty,,¢.=1aswell.ThemapJisdefinedtobethe composite j10voftwo involutions y«and v,which aredetermined bythe following: (i)interchangesS*andS~,andmapsVtoitself,with4(s)=v,-sfors.€S*; lv)=240,v1v0—vforveV. (ii)vinterchanges Vand S”,maps S*toitself, with v(0)=vs,forveV; W(3)=245,51)5-5,—slorseS* ‘Show that thisJsatisfies theasserted properties Exercise 20.51%, Inthisalgebraic form, triality canbeexpressed bytheasser- tionthatthere isanautomorphism jofSpingC oforder 3compatible with J, ie,such thatforallxeSpingC, thefollowing diagrams commute: §203. Sping€ andTriality a5 ys +.5 7 ne Ifj':80gC-+509Cisthemapinducedbyj,thefactthatjiscompatiblewith thetrilinear form ®(cf.Exercise 20.49) translates tothe“local triality” ‘equation (Xe, 5,0) +Oe, ¥5,0)+0, 5,Zt)=0 forX€809, ¥=j'(X),Z =j'(¥). PART IV LIE THEORY ‘Thepurposeofthisfinalpartofthebookisthreefold.Firstofall,wewanttocompletetheprogramstatedintheintroduction toPart Il.Wehave completed thefirst twosteps ofthisprogram, showing in Part IThow theanalysisofrepresentations ofLiegroupscouldbereducedto thestudy ofrepresentations ofcomplex Liealgebras, ofwhich themost important arethesemisimple; and carrying outinPart I1fsuch ananalysis fortheclassical Liealgebras s1,C, sp24C, and#0,C. Tofinish thestory, wewantnowtotranslateouranswersbackintothetermsoftheoriginalproblem. {Inparticular, wewant todeal with representations ofLiegroups aswell as Liealgebras, andrealgroups andalgebras aswell ascomplex. The passage back togroups isdescribed inLecture21,andtheanalysisoftherealcasein Lecture 26. ‘AnothergoalofthisPartistoestablishaframeworkforsomeoftheresults ofthepreceding lectures—to describe thegeneral theory ofseinisimple LiealgebrasandLiegroups.Thekeypointhereistheintroduction oftheDynkindiagram anditsuseinclassifying allsemisimple Liealgebras over C.From conepoint ofview, theimpact oftheclassification theorem isnotgreat: itjust tells usthat wehave infactalready analyzed allbutfiveofthesimple Lie algebras inexistence, Beyond that, however, itprovides apicture and @ language forthedescription ofthegeneral Liealgebra. This both yields a description ofthefiveremaining simple Liealgebras andallows ustogive uniform descriptions ofassociated objects: forexample, thecompact homo- ‘geneous spaces associated tosimple Liegroups, orthecharacters oftheirrepresentations. Theclassification theoryofsemisimple LiealgebrasisgiveninLecture 21;thedescription inthese terms oftheir representations and characters isgiven inLecture 23.The fiveexceptional simple Liealgebras,whoseexistenceisrevealedfromtheDynkindiagrams, arestudiedinLecture 318 Pot IV. LieTheory 22;wegive afairly detailed account ofone ofthem (g3), with only brief descriptions oftheothers. Third, allthisgeneral theory makes itpossible toanswer themain out-standingproblemleftoverfromPartIII:adescription ofthemultiplicities of theweights intheirreducible representations ofthesimple Liealgebras. WesiveinLectures24and25anumberofformulas forthese multiplicities, This, itshould besaid, represents insome ways ashift instyle. Inthe previous lectures wewould typically analyze special cases firstanddeduce general patterns from these cases; here, forexample, the Weyl character formula isstated andproved ingeneral, then specialized (0thevariousindividual cases(thisistheapproachmoreoftentakenintheliteratureonthe subject). Insome ways, thisisafourth goal ofPart IV:toprovide abridge between thenaive exploration ofLietheory undertaken inParts Ifand III, andthemore general theory readers willindelsewhere when they pursue the subject further. Finally, weshould repeat here thedisclaimer made inthePreface, This part ofthebook, totheextent that itissuccessful, will introduce thereader totherichandvaried world ofLietheory; butitcertainly undertakes no serious exploration ofthat world. Wedonot, forexample, touch onsuchbasicconstructions astheuniversalenveloping algebra,Vermamodules,Titsbuildings; and wedonoteven hint atthefascinating subject of(infinite- dimensional) unitary representations. The reader isencouraged tosampletheseandothertopics,aswellasthoseinclydedhere,according tobackgroundand interest. LECTURE 21 The Classification ofComplex Simple LieAlgebras Tnthefirst section ofthis lecture weintroduce theDynikn diagram associated tosenisimpleLiealgebrag,Thissanamazinglyefficientwayofconveyingthestructureofgiteasimplediagramthatnotonlydeterminesguptoisomorphism intheory,butinpracticeexhibitsmanyofthepropertiesofThemainuseofDyakindiagrams fnthis lecture, however, wll betoprovide aframework forthe basc classification theorem, which says that with exactly fiveexceptions theLiealgebras discustedsofar inthese lectures areallthesimple Liealgebras. Todothis in§21.2 weshow how to listalldiagrams that arise from semisimple Liealgebras. In§21.3 weshow how to recoversuchaLiealgebrafromthedataofitsdiagram, completing theproofofthe classification theorem. Allthree sections arecompletely elementary, though §21.3 gets Alitle complicated; may beuseful toread itinconjunction with §221, where the Process deseibediscarid out indetail frtheexceptional algebra gy.(Note thatneither§21.3of§22.1isaprerequisite for§22.3,whereanotherdescription ofgywillbbegiven.) $21.1:Dynkindiagramsassociatedtosemisimple Liealgebras$212: ClassifyingDynkindiagrams 212: RecoveringaLiealgebrafromitsDynkindiagram §21.1. Dynkin Diagrams Associated toSemisimple LieAlgebras Forthefollowing, wewillletgbeasemisimple Liealgebra;asusual,aCartansubalgebra hofgwillbefixedthroughout. Aswehaveseen,therootsRofg‘span arealsubspace ofh*onwhich theKilling form ispositive definite. We denote thisEuclidean space here byE,andtheKilling form onEsimply by x0 2TheClasifcation ofComplex Simple LieAlgebras (.,insteadofBC,)Thegeometry ofhowRsitsinEisveryrigid,asindicated bythepictureswehaveseenfortheclasicalLiealgebras.Inthissectionwe willclassify thepossible configurations, uptorotation andmultiplication by 2positive scalar inE.Inthenext section wewill eethat this geometry completely determines theLiealgebra “Thefollowing fourproperties oftherootsystem areallthatareneeded: (1)Risa fot setspanning € (Q)weR= ae R,butk-aisnotinRifkisanyrealmmberotherthan+1. (G) ForeR,thereflectionW,inthehyperplaneamapsRolise (4)Fora,feR,therealnumber 7) ye =12)Ga) (san integer. Except perhaps forthe second part of(2),these properties have been seen inLecture 4.Forexample, 4)isCorollary 1429, Note thatng,=B(H,), and WB) =B=ryt eu) For(2consider therepresentation i=(De oftheLiealgebras, =#,C. Note that allthenonzero factors but)=:gare onedimensional. Wemay assume aisthesmallest nonzero root that appears Inthestring. Now, decompose jasan«module: ine@i. Bythehypothesis thataisthesmallest nonzero ootthatappears inthestring, iisarepresentation of, having noeigenspace with eigenvalue |or2for fy. Iefollows that must betrivial 12,gy,=(0)for k#Or +L‘AnysetRofelementsinaEuclideanspaceEsatisfyingconditions(1)to(4)maybecalledan(abstract)rotsysteProperty ()puts very strong restrictions onthegeometry oftheroots. If ‘Distheangle between aandf,vehave MnDoon. 12) Inparticular, Appa =4008%) a3) isaninteger between 0and4.The case when thisinteger is4occurs when cos(9)= 1,1. p=ba.Omitting thistrivial case, theonly possibilitiesae therefore those given inthe following table. Here wehave ordered the(wo roots0thatIAL2FathoFtl=ny $21.1. Dynkin Diagrams Associated toSemisimple LieAlgebras 321 ‘Table 21.4 coi) 32 J22 12 0 -12 -fyp -/3p a a6 ai nfn/2Inf} Awl ‘Sn/6reSoeOeeraeeea Bow wetet fw Tnother words, therelation ofanytworootsaandfisoneof a sus ‘Thedimensionn=dimgE=dimchiscalledtherank(oftheLiealgebra, ortheroot system). Itiseasy tofindallthose ofsmallest ranks. Aswewrite them down, wewilllabel them bythelabels (A,), (B,), ..-that have become standard. Rank 1.Theonly possibility is eee whichistherootsystemofs1,C. Rank 2.Note firstthat byProperty (3),theangle between tworoots must be theame foranypalrofedjacrat root Ina two-dimensional root sytem. As ‘wewillsee,anyofthefour angles x/2,n/3,n/4,andn/6canoccur; once this angle isspecified therelative lengthsoftherootsaredeterminedbyProperty (4),exceptinthecaseofright angles. Thus, uptoscalars there areexactly four rootsystems ofdimension two. First wehave thecase ©=/2, ‘(ayeanmts which istheroot system ofs1;€ xsl,€ =#04.(Ingeneral,theorthogonal directsumoftworootsystemsisarootsystem; an 21,TheClassification ofComplexSimpleLieAlgebras ‘root system that isnotsuch asuum iscalled irreducible. Our task willbe toclassify allirreducible root systems.)Theotherrootsystentsofrank2are therootsystemofslsC; theroot system ofs04€ =sp4€; and .MK AlthoughwehavenotyetseenaLiealgebrawiththisrootsystem,wewilleethat there isone. Exercise 21.5, Show that these arealltheroot systems ofrank 2. Exercise21.6.Showthatasemisimple Liealgebraissimpleifandonlyiftsroot system isirreducible. Rank 3.Besides thedirect sumsof(A,)withoneofthoseofrank2,wehave theirreducible root systems wehave seen; wedraw only dots attheendsof thevectors, theorigins being inthecenters ofthereference cubes: ; {BLL Dyokin Diagrams Associated toSemisimple LieAlgebras m —s | + ay \ ! oo a” - which istherootsystem ofsl,C=s0gC; — a - ‘* Ei i therootsystem ofs0,C; therootsystem ofsp¢C. Exercise 21.7.Show thatthere arenoother rootsystems ofrank3. Wecan further reduce thedata ofarootsystembyintroducing asubsetof theroots, called thesimple roots. First, choose asinLecture 14adirection 304 21,TheClassification ofComplexSimpleLieAlgebras LER,sothatR=R*UR”isadisjointunionofpositiveandnegativeroots.Callapositiverootsimplefitisnotthesumoftwo other positive roots. For theclassical Liealgebras, keeping thenotations and conventions of Lectures 15-20, thesimple roots are (A) Shae©by=bayba=bayeybgt~bybg—nots (B)—s0aq44© by hayba—baggy beat =baybe (Cy) Bagby =hayby=bay enepbaa ~boyDo (Dy) 802 Ly~hagha~Lyeybt~baybt+by Exercise 21.8. Verify thislist,andfindtwosimple roots for(G). Wenextdeduce afewconsequences ofproperties (1)-(4), which indicate hhow strong these axioms are.They willbeused inthepresent classificationof abstract systems, aswell asin thefollowing section. (5)IfafarerootswithB#+a,thenthea-stringthroughfiLe.therootsof theform B=paA(p=NsA=4,8,B+B+20,5B+QO thasatmost four inastring, Le. p+q<3; inaddition,p~q=Mya. Indeed, since W.(A +qa)=f)— pa,and WB+92)=(B—ny)—Qa, ‘wemust havep=nig,+q,Whichisthesecondequality.Forthefirst,wemay {akep=0,and then q=—M, which wehave seen isaninteger nolargetthanthree.Asaconsequence of(5)wehave (6)Supposea,arerootswithf#a.Then(fa)>0a—fiisaroot, (fa)<0->a+fisaroot IF(B,2)=0,then a—anda+faresimultaneously rootsornonroots.(1)WfcandParedistinctsimpleroots,thena—ftandfl—aarenotroots. This follows from thedefinition ofsimple, since from theequation a= fi+(a— fi),«—ficannotbeinR*,andsimilarly—(«—)=—« cannotbeinR*.From(6)and(7)wededucethat(,f)<0,ie, (8)Theangle between twodistinct simple roots cannot beacute. (0)Thesimple roots arelinearly independent. This follows from (8)by Exercise 21.9*. Ifasetofvectors liesonone side ofahyperplane, withal ‘mutual angles atleast 90°, show that they must belinearly independent. $21.2. Classifying Dynkin Diagrams 2s (10) There areprecisely nsimple roots. Each positive root can bewritten uniquely asanon-negative integral linear combination ofsimple roots. Since Rspans E,thefirststatement follows from (9),asdoes theuniqueness ofthe second statement. The factthat anypositive root canbewritten asapositivesumofsimple roots follows readily from thedefinition, forifawere a positive root with minimal (a)that could notbesowritten, then aisnot simple, soa=f+y,with fandypositive roots with I(A) IQ)<l(a. Note thatasanimmediate corollaryof(10)itfollowsthatnorootisalinear combination ofthesimplerootsa,withcoefficientsofmixedsign.Forexample, (Dis just aspecial case ofthis. ‘TheDynkin diagram oftheroot system isdrawn bydrawing onenode Oforeachsimplerootandjoiningtwonodesbyanumberoflinesdepending.‘ontheangle 9between them: rotines 00teem onetne oo tons woes SED itOeae eels=CESED HOSs ‘When there isoneTine, theroots have thesame length; iftwoorthree lines, anarrow isdrawn pointing from thelonger totheshorter foot. Exercise21.10.ShowthatarootsystemisirreducibleifandonlyifitsDynkin diagram isconnected. WewillseelaterthattheDynkindiagramofarootsystemisindependent ofthechoice ofdirection, i,ofthedecomposition of into R*andR~ §21.2. Classifying Dynkin Diagrams The wonderful thing about Dynkin diagrams isthat from this very simplepictureonecanreconstruct theentireLiealgebrafromwhichitcame,Wewillseethisinthefollowingsection;fornow,weaskthecomplementary questionofwhichdiagramsarisefromLiealgebras.Ourgoalisthefollowingclassifica-tiontheorem, which isaresult inpure Euclidean geometry. (The subscriptsonthelabels(A,),...arethenumberofnodes.) ‘Theorem 21.11. TheDynkin diagramsofirreducible rootsystemsareprecisely: us 3.TheCieonCompeSigeLiAst @ oo oD en ) o-oo te) oo o o—asD—o @ oxen The first four arethose belonging totheclassical series wehave been sadn (An) Shor (By) $0a_41€ C) Pf Dy) 62,C ‘The restrictionsonmintheseseriesaretoavoidrepeats,aswellasdegenerate caine teSagres tofal ae oneness ee cn When m= our ofthe diagrams become one noe. Te ease)degensnsansenotseminmpl wetecnncenca(y) =1h)= (A,)correspond totheisomorphisms sp,C=s0,C=8l,C oO. £212, Classifying Dynkin Diagrams By Forn=2,(Dz)=(Ay)x(A;)consistsoftwodisjointnodes,correspond-ingtotheisomorphism s0,Ces,CxslC O oO. ‘Thecoincidence (C,)=(B,)corresponds totheisomorphism sp,€ &$046 OED =CHD} Forn=3,thefactthat (D5) =(A) reflects theisomorphism wceae<=0~o—0 PRoor oFTHETHEOREM. Our desert-island reader would findthisapleasant pastime. Forexample,iftherearetwosimplerootswithangle52/6,theplane ofthese roots must contain theG,configuration of12roots. Itisnothard toseethatonecannotaddanotherrootthatisnotperpendicular tothisplane,without someofthe 12angles and lengths being wrong. This shows that (G2) istheonlyconnected diagram containing atripleline,Attheriskofspoiling‘yourfun,wegivethegeneralproofofaslightlystrongerresult.Infact,theanglesalonedetermine thepossible diagrams, Suchdiagrams, without thearrows (oindicate relative lengths, areoften called Coxeter diagrams (orCoxeter graphs). Define adiagram ofnnodes, with each pair connected by0,1,2,or3lines,tobeadmissible iftherearenindependent unit ‘vectorse,,...,¢,inaEuclideanspace€withtheanglebetweene,ande,being 1/2,2n/3,3n/4,OrSn/6,accordingasthenumberoflinesbetweencorrespond- ingnodes is0,{,2,or3.The claim isthatthe diagramsoftheaboveDynkin diagrams, ignoring thearrows, aretheonly connected admissible diagrams. ‘Note that (e,€) =0,-1/2, -/272, or—/372, 21.12) according asthenumber oflines between them is0,1,2,or3;equivalently, 4(e,,¢)"=numberoflinesbetweene,and¢,.(21.13) Thestepsoftheproofareasfollows: (0Any subdiagram ofanadmissible diagram, obtained byremoving some odes and alllines tothem, wil also beadmissible. (ii)Thereareatmostn—1pairsofnodesthatareconnected bylines.Thediagram hasnocycles (loops). Indeed, if;and¢,areconnected, 2(¢, )<—1,and O<FeLelan42F (ene whichprovesthefirststatement of(ii). Thesecond follows from thefirstand(i). we 21,TheClassfiation ofComplexSimpleLiAlgebras (iii) Nonode hasmore than three lines toit. By(i),wemay assume that e,isconnected toeach oftheother nodes; by Gi,noother nodes areconnected toeach other. We must show that Lh-24(e,,€))? <4.Since e3,...,¢,areperpendicular unitvectors,ande,is notin their span, 1=(,,¢,) >dfee), asrequired, (iv)Inanadmissible diagram, anystring ofnodes connected toeach other byoneline, with none buttheends ofthestring connected toanyother nodes, canbecollapsed toonenode, andresulting diagram remains admissible: ae 2 ao ° © o co fey, ...,¢aretheunit vectors corresponding tothestring ofnodes, then ese, 4°" +e,isaunitveetor,since (ee) =r+ley, e2)+(ea,es)4° +(C1 EN) =r-(=). Moreover, e’satisfies thesame conditions with respect totheother vectors since (e’,¢))iseither (e,,¢))ore,,¢))-Nowwecanruleouttheotheradmissibleconnecteddiagramsnotonout list.First,from(iii)weseethatthediagram(G,)hastheonlytripleedge.Next, there cannot betwodouble lines, orwecould find asubdiagram oftheform: a=p—o—.... —a==p andthen collapsethemiddletogetC—— contradicting (ii.Similarly therecanbeatmostonetriplenode,i.c.,anodewithsinglelinestothreeothernodes, by $212. Classifying Dynkin Diagrams 39 Bythesame reasoning, there cannot beatriple node together with adouble line: Tofinish thecase with double lines, wemust simply verify that o—a-=p—o—o cs a) isnot admissible. Consider general vectorsv=aye,+0302,andw'=aye,+ Ae, +ayes. Wehave WI?=aj+a} —aya,, wl)? =a}+a}+a3—asay —agas, and(0,)=~a,a,/,/2, Wewant tochoose vandwtocontradict the Cauchy ~Schwarz inequality (0,w)?<Jof*l}wl?. Forthiswewant |a,\/foll and[a,//Iw tobeaslarge aspossible. Exercise21.14.Showthatthesemaximaareachievedbytakinga,=2a,anda= 3a, 0,=2a. Infact,v=€,+2es,w=3ey+2ey+edogivethecontradictory (ow)? =18, fo? =3, and fl? =6. Finally,wemustshowthatthestringscomingoutfromatriplenodecannotbelongerthatthosespecifiedintypes(D,)(E),(E;),0F (E).First,weruleout Consider thethree perpendicular unit vectors: w=Reptes0=(egtes/3,w=(egte/3. Thenasin(li),sincee,isnotinthespanofthem, 330 21.TheClasiiation ofComplexSimpleLieAlgebras T=Hes?>(ey,u?+(er0?+(erWP=1/3+13+1B=1, acontradiction. Exercise21.15*,Similarly,ruleout o-o-o-bo-o-0 and te ee (The lastfewargumentscanbeamalgamated, byshowingthatifthelegsfrom atriple node have lengths p,4,and r,then t/p-+ 1/q+I/rmust begreater than 1)Thisfinishestheproofofthetheorem. a §21.3. Recovering aLieAlgebra from Its Dynkin Diagram Inthis section wewill complete theclassification theorem forsimple LiealgebrasbyshowinghowonemayrecoverasimpleLiealgebrafromthedataofitsDynkin diagram. This will proceed intwo stages: first, wewillseehow {0reconstruct aroot system from itsDynkin diagram (which apriori only {ells ustheconfiguration ofthesimple roots). Secondly, wewill show how todescribe theentire Liealgebra interms ofitsroot system. (Inthenext lecturewewilldoallthisexplicitly,byhand,andindependently ofthegenerdiscussion here,forthesimplestexceptional case(G}aswehavenoted,thereader may find ituseful towork through §22.1 before orwhile reading the general story described here)Tobeginwith,torecovertherootsystemfromtheDynkindiagram,let,14,bethesimplerootscorresponding tothenodesofaconnected Dynkindiagram, We must show which non-negative integral linear combinations Sma, areroots. CallFm,thelevelof¥may.Thoseoflevelonearethesimp ‘Toots. Forlevel two, weseefrom Property (2)that no2aisaroot, andby {P13 RecoveringaLieAlgebrafromItsDynkinDiagram a Property (6)that«+a,isaroot precisely when (a,04)<0,ie,when the corresponding nodes arejoined byaline.‘Supposeweknowallpositiverootsoflevelatmostm,andletf=Sima, bbeanypositiverootoflevelm.Wenextdetermine foreachsimpleroota=a),whether f+«isalsoaroot. Look atthea-string through fi: B~ pesos Proves B+OR \Welnow pbyinduction (10root isalinear combination ofthesimple roots 4,withcoefficients ofmixed sign, sop<myandff—pais apositive root). By Property (S).q°= P—Me:80f+aisaot exactly when (8.0)_ >tye=28 oF ngPte Baa) By Inelffect, theadditional roots wewillfindinthisway arethose obtained by reflecting aknown positive root inthehyperplane perpendicular toasimple roota,(andfillinginthestringifnecessary). Tofinish theproof,wemustshowthatwegetallthepositiverootsinthis way.Thiswillfollowoncefromthefactthatanypositiverootoflevelm+1 fanbewritten inatleast one way asasum ofa poitive root oflevel mand asimple root. Ify=Fora, haslevel m+|,from 0< =Lua some (7,1) must bepostive, with r,> 0.Byproperty (6hy~a8 aroot, as required.Bywayofexample, considertherank2rootsystems.Inthecaseofsly,westart with apairofsimplerootsa,@,withn,,,.,=1,ie,atanangleof2/3; asalways, weknow that =a,+9, isaroot a5wel. a) peate, Vaa Ontheother hand, since ff—2a,=a2—ayisnotaroot, f+a, cannot be either, andlikewise +a,isnot;sowehaveallthepositiveroots. Inthecaseofsp4C,wehavetwosimplerootsa,anda,atanangleof3n/4; intermsofanorthonormal basisL,andL,thesemaybetakentobeL,and 1, Ly,respectively. % ata, Jota, INVA« ey 21.TheClasifcation ofComplerSimpleLieAlgebras We thenseethatinadditiontot=+a,thesum+2,=2m,isa root—itisjusttherectionofa,Intheplaneperpendicular (0@,—-bulB+a,=a,+2a,and3a,+aarenotbecausea,—a,anda,—a,arenotrespectively (alternatively, wecould note that they would form inadmissible ingles witha, and arespectively) Finally, inthecaseof(G,),wehavetwosimpleroots«,,a,atanangleof ‘5n/6, which intermsofanorthonormal basisforEmaybetakentobeL,and (—3L, +/3L)/2 respectively ”sae~ Reflecting «,intheplane perpendicular toa,yields astring ofroots a,+a, a,+2a,and a,+343. Moreover, reflecting thelastofthese intheplane perpendicular toa,yields onemore root, 2a,+3ay. Finally, these areallthe positive roots, giving ustheroot system forthediagram (G,). We state here the results ofapplying this process totheexceptional diagrams (F,),(Es),(E,),and(E,)(inaddition to(G,)).Ineachcase,Ly,.... LL,isanorthogonal basisforE,thesimplerootsa,canbetakentobeasfollows, andthecorresponding root systems aregiven: 3,v3 (G.)ayehy aye 5h tb R={lnVitethy+Bu,3b,+f} (Gz) thus has6postive roots. (F,) a,=L,—-Ly, a=Ly-ly, as=ly, gwbtba=babe= poe; + Ly+bytby+Ly RP=[LUflatLheUla—bjfee Habh Inparticular, (F) has24postive roots (6yeh abstVisgetthy {21.. Recovering LieAlgebrafomIlsDynkinDiagram m ayeby—byy=byl as=Ly— Ly agebyLys RYm(a+ bhicsesU(l=Lyhyeresgy{tbatbatbtLabbs+Vibe2 numberofminussignseven (E,)has36positive roots. &)aebyabaninsbetSilo,ay=hy+Ly Bye La— by abybytpbyby ageLsy—Ly, y=be—bs; RE=(Let LyeseoUla~Lyhyctce(2b) vfthth Bibles iby . z orem ee ‘Thus, (E,) has63positive roots. Ly=y= bythy G) gation thy ay=hy—byageLby—Lz, as=by—Ly agaLs—Ly, aebe—Ls, ty=Ly—Le. RY=(Ly+LyhrcsenUla~Lyhyaten vutL, tL, te thy+by - 2 Samberstminastnteven” {E,)has120positiveroots.For(G,) and (Fy) thesimple roots arelisted inorder reading from letto fightintheir Dynkin diagrams: oo o—a0—o sintheclassical series (A,)-(D,). For(Ey), thenumbering is %) Gs 6 as Gy a oF a4 21,TheClassification ofComplex Simple LieAlgebras while those for(E,) and (E,) areobtained byremoving thelastone ortwo nodes. Note that, given theroot system of(Ey), wecan find theroot systemof(E,)or(Eg)bytakingthesubspacespannedbythefirstsevenorsixsimple roots. Exercise 21.16*. (8)Verify theabove lists ofroots.(b)Ineachcase,calculatethecorresponding fundamental weights Exercise 21.17%. Show that notwo ofthe root systems of(A,)-(Ey) ateisomorphic, anddeducethattheDynkindiagramofarootsystemisindepen-dent ofchoice ofpositive roots ‘Amore satisfying reason forthelastfactistheobservation that any(wo choices ofpositive roots difler byanclement oftheWeyl group—the group generated byreflections W,inthesimple roots. This canbeseen directly for ‘each ofthediagrams (A,)-(E,); forageneral proof that twochoices differ by anelement oftheWeyl group, seeProposition D.29‘Weshouldmentionhereanotherwayofconveying thedataofaDyakiadiagram. This issimply themxmmatrix ofintegers (my=Maa). where Wetakem=2;itiscalledtheCartanmatrixoftheDynkindiagram(oroftheLiealgebra).Thus,forexample,theCartanmatrixof(A)is 2-1 0: 0 -1 2-1 0 + + 0 o-1 2-1 + - 0 0 0 s-t 2-1 0 0 a ‘Thesematricespopupremarkably often,in@varietyofseemingly unrelated areas ofmathematics. They willnotplayamajor roleinthepresent text,bal thereader hasprobably encountered them already inoneform ornoth andwillprobably dosoagain. Exercise21.18%.ComputetheCartanmatrix,anditsdeterminant, foreadDynkin diagram. ‘Theneattaskistoseehowtherootsystemdetermines theLiealgebra.We‘concentrate ontheuniqueness, sincethereareotherwaystoseetheexiste indeed, forallbutthefiveexceptions wehave already seen theLiealge‘Wewilldescribeseveralapproaches tothisproblem,startingwithastaighforward and computational method and finishing with aslick butab approach, B13. Recovering aLieAlgebra from ItsDyakin Diagram Assume asbefore that gis&simple Liealgebra, with achosen Casubalgebra banddecomposition oftherootsRintopositiveandnegsroots;leta,»dybethesimpleroots.TheDynkindiagraminformatictheknowledge of(%,a)fralli#j,LetH,=H,,bethecorresponding t(of5,definedbytherulewehaveseeninLecture14:if(7;}isthebcorresponding viatheKilling form to{a,), setH,=27;/(a, 4) Choose anynonzero eleinent X,intheroot space g,,for1<i< m1determines elementsYing_.,suchthat[Xj¥.]=Hy.Weclaimfirstthattt Inelements {1,X,,Yi}generate gasaLiealgebra. This follows from Chaim 21.19. Ifaf,anda+Bareroots,then[ay84]=Bary Proor,Againlookatthea-stringthroughgp,ie,Paezysse:Thisisireducible representation ofs,=sl,C,sinceallthetermsareonedimensio(thisfollowsfromthefactthatnofi+kaecanbezero,giventhatf#:ta).} 10Wif(a49p]=0,azo9pr4aWOuld beanontrivialsubrepresentation, Foreach positive root p,wehave seen thatcanwritefasasumofsim foots f=a,++a,suchthateachofthesumsa,+++a,is8£0 1's <r. If'we choose such &presentation foreach f,and set Xp OM KygeenMigMeo] and MyOs geeMi Hdd then thecollection {Ho 1<i<n: Xp,YyBER") Qt2 forms abasis forg,Note thatiffisnotsitnple, there isnoreason toexpe {Uiy, %]tobethedistinguished element Hyin. “We want toshow thatthemultiplication table forthese basis elements sgompletely determined bytheDynkin diagram. Themaindifficulty isthattt {isinofthse oottheaboveexpremion formayntbeania ‘Forexample, suppose B=(ay+0)+ay=(02+43)+04, sitha,+a,anda+ayroots,Wemustcompare(Xs,(Xz,XJ}wit ‘BG,L%5,X71].Infact,theymustbenegativesofeachother.For,byJacob Whave BouO43,A071=—0%,(2,0]~2X2XX31=—LEOXMD] ingthat [X,, Xj] =0since a+aycannot bearoot, eg, bystep (i)0 preceding section.ForanysequenceI=(iy,....ij) 1<ijSn.set 336 21.TheClassification ofComplex Simple LieAlgebras ayy tebay, Xp=XsXegrees[XieXi Ya=DisTiareMayHidTI Call1admissibleifeachpartialsuma,,+--++4,isaroot,1<8<r;note that 1isadmissible exactly when X;isnotzero, Lemma 21.21. 1fandJaretwoadmissiblesequencesforwhichay=a,then thereisanonzerorationalnumberqdeterminedbyI,J,andtheDynkindiagran, such that X,=q°X,. Proor. Letk=i,bethelastentry inI.Ifj,=kaswel, theresult follows by induction onr.Wereduce thegeneral case tothiscase, bymaneuvering to replace j,byk.Wehave first X=4°,OhXT with qyanonzero rational number depending only onJ,k,and theDynkia diagram, since a,~ay=a~aisaroot; thepoint isthat weknow how s.,&sl,actsontheay-string through a,assoonasweknowthelengthofthe string, and thisisDynkin diagram information. Next, letsbethelargest integer such thatj,=k.Then TheaD=DXp0oo(MaesOlasOMeXII)Js whereK=(jas-+-yfdbsince(%,[Xi21]=(Xn[Yie21]when! k.Finally, Uh,Xn XJ =42°Xe withq2@nonzero rational number depending only onK,k,andtheDynkia’diagram,sincea+oisaroot.Combining thesethreeequations, weget Xp 9192 XsXi [Xia Kad De which sulfices since thesequence fortheterm ontheright ends inthesemé integer kas1. is} Proposition 21.22.Thebracketofanytwobasiselementsin(21.20)isaresin multiple ofanother basis element, that multiple determined from theDyn diagram. PoorThiscetforbractsofanH,withanybasscement.Lemna2.3 handlesbracketsoftheform[X;,X,],andthoseinvolvingonlyY's similar. Forbrackets (¥,,X,], itsufices inductively tocompute [%,X] arational multiple ofsome Xx,with Kshorter than J(orofHyiJhas: term}; butthiswasworked outintheproof ofthelemma. Exercise 21.23* (i)Show thatin(G,) each positive root canbewritten inoalf oneway asasum ofsimple roots, uptotheorder ofthefirst tworoot BLA. Recovering aLieAlgebra from ItsDynkin Diagram : (ii)Work outthemultiplication table from theDynkin diagram. (ii)Ver: thattheresult isindeed aLiealgebra, which is(visibly) simple. Thisexercise willbeworked outindetailtostartthenextlecture.Ofcour: thereisnothingbutlackoftimetokeepusfromverifying thattheotherfo exceptional Dynkin diagrams dolead, bythesame prescription, tohonest L algebras, butdoing itbyhand getspretty laborious, andwewilldescribe son oftheothermethods available.The fact that themultiplication table can bedefined with rational coef cients becomes important when one wants toreduce them modulo prin numbers, which wewillnotdiscuss here. Thefactthattheycanbetaken t bereal, ontheother hand, willcome uplater, when wediscuss realforms complex Liealgebras and groups. There isamoregeneralandelegantwaytoproceed,givenbySerre[Se3 ‘Writenyinplace ofn,,,,.Form thefreeLiealgebra ongenerators HsoeeyHayXpsooesXaYoseoosFs ic,form thefree(tensor) algebra with thisbasis, anddivide modulo byth relations (4,B]+[B,A]=0andtheJacobi relation. Then take thisfreeLi algebra, anddivide bytherelations CH,A]=O(a CXHI= Mall X,Y OCA I: Ui,Xp)=myX(0llHeCHYD=ryYall and,for alli#j, (4%I=0 OH=0 itmy=o; COL 410 LK. WIT=0 itmy= —1 (XO OXKI=0,OOO, WTO itty=~2. Do1%, 2%0%47 =0 OOK DL. HIT=0ifmy=—3 Exercise21.24.Verifythatifonestartswithasemisimple Liealgebrawitha'BvenDynkin diagram, theaboveequations musthold. Serre shows ({Se3, Chap. VIApp.], cf.[Hut §18)) that theresulting Lie _agcbra isinite-dimensional semisimple Liealgebra, withCartan subalgebra 1 tedbyH,,..., H,andgiven root system. Inparticular, thisincludes a‘roototheexistenceofallthesimpleLiealgebras. Here isathird approach touniqueness. Supposegandg},withgivenCartan phar§and ¥¥,and choice ofpositive roots, have isomorphic root‘ystems.Thereisanisomorphism b->1,takingcorresponding 1,toHj.{Choose arbitrarily nonzero vectors X;andX;intherootspaces ofgand ‘Sortesponding tothesimpleroots. 338 21.TheClassification ofComplexSimpleLieAlgebras ‘Claim 21.25. There tsaunique isomorphism from gtog'extending theiso ‘morphism of§with W,andmapping X;toX;forall ProoF. The uniqueness oftheisomorphism iseasy: theresulting map is determined onthe¥,bysl,considerations, and the1,X;,and ¥,generate8 Fortheeristence oftheisomorphism consider thesubalgebra gofg®9 generated by =HOH}, X,=X,@Xj, andK=¥@Y.1suffices toprovethatthe(woprojections from§(ogandg'areisomorphisms. Thekernel ofthesecond projection ist@0,where tisanideal in9,Since gissimple, iseither 0,asrequired, or=g,Inthelatter case, wemust have §=9@8 Toseethat this isimpossible, consider amaximal positive root A, take nonzero vectors X,,Xjinthecorresponding root spaces, and setX,=Xp@Xp,ahighestweightvectoringLetWbethesubspaceof§‘obtainedbysuccessively applyingallfs.ThenWisapropersubspaceofsince itsweight space W,corresponding {0fis one dimensional. Bythe argument wehave seen several times, §preserves W.Now ifj=9@q, [email protected],@0tobelongtomakingW,twodimensional again. a Tofinish thisstory, weshould show that thesimple Liealgebras corre sponding totwo different Dynkin diagrams cannot beisomorphic, ic,that thetwochoices made ingoing from asemisimple Liealgebra toDynkin diagram donotchange theanswer. The general facts are: (1)Any two Cartan subalgebras ofasemisimple Liealgebra areconjugate ie,there isaninner automorphism byanelement inthecorresponding adjointgroup,whichtakesoneintotheother. (2)Anytwodecompositions ofarootsystemintopositiveandnegativerootsdiffer byanelement oftheWeyl group. ‘These arestandard facts which areproved inAppendix D.Both statementsaresubsumed inthefactthatanytwoBorelsubalgebras ofasemisimple Litalgebraareconjugate, aBorelsubalgebra beingthesubspacespannedbytheCartan subalgebra andtheroot spaces g,forpositive &Forthose reader who crave logical completeness butdonotwant (0gothrough somuch general theory, weobserve that most possible coincidences canberuled out bystch simple considerations ascomputing dimensions, andothers canbe ruled outbysimple adhoc methods, f.Exercise 21.17 Finally, wemustalsoprovethe“existence theorem”: thatthereisasimpLiealgebra foreach Dynkin diagram, Serre’s theorem quoted above gives unified proof ofexistence. Butwehaveseenandstudied theLiealgebrasot theclassicaleases(A,)-(D,), anditismoreinkeepingwiththespiritofthealectures toatleast trytoseethefiveexceptions explicitly. This isthe subjed ofthenext lecture, LECTURE 22 §2and Other Exceptional LieAlgebras “Thislectureimainlyaboutg,Withjustenoughdscusionofthealgebraicconstrvc-sonsoftheotherexceptionalLialgebrastogivethereaderasenseoftheircomplexity.dibeingonly{4dimensional, diferent:wecanreasonablycaryoutinpracticetheprocessdescribedin§21.3toarriveat anexplicitdescriptionofthealgebrabyspecifying basis andallpairwise products; wedothisin§22.1 andverify in§22.2 thattheresult really isaLiealgebra.In§22.3weanalyzetherepresentations ofq3,andarriveinparticular atanother description ofgz:itisthealgebraofendomorphisms ofseven-dimensional vector space preserving ageneral trilinear form, (Note that §22.3 maybereadindependently ofeither§22.1,§21.2,or§21.3)Finally,inthefourthsection ‘rewill sketch some ofthemore abstract (ie, coordinate free) approaches tothe ‘onstruction ofthefive exceptional Liealgebras. While thefirst two sections are completely elementary, theconstructions given in§22.4 involve some fairly serious algebra 22.1:Construction ofg,fromitsDynkindiagram222: Veriying thatquit Liealgebra 223: Representation theory ofg2 {P24 Algebra constructions ofthe exceptional Liealgebras $22.1. Construction ofg,from ItsDynkin Diagram ‘Inthis section wewillearry outexplicitly theprocess described inthe preceding ‘ection fortheDynkin diagram (G,),constructing inthiswayaLiealgebra ‘awith diagram (G,) (and inparticular proving itsexistence). “The firststep istofindtheroot system from theDynkin diagram. Inthe caseofg thisisimmediate; wemay draw theroot system Rch associated 338 21,TheClassification ofComplexSimpleLieAlgebras Claim 21.25. There isaunique isomorphism from g(0'extending theiso ‘morphism of6with W,andmapping X,toX;foral PRooF. The uniqueness oftheisomorphism iseasy: theresulting map is determined onthe¥;bysl,considerations, and the1%,X;,and Y,generateg. For theexistence oftheisomorphism consider thesubalgebra §ofgagenerated byA,=H,@Hi,X=X,@Xi,[email protected] from§toandg'areisomorphisms. Thekernel(ofthesecondprojection is@0,wheretisanidealin,Sincegissimple,iseither 0,asrequired, orf=g,Inthelatter case, wemust have §=g@ Toseethat this isimpossible, consider amaximal positive root f, take nonzero vectors X,,Xjinthecorresponding root spaces, andset X,=X,@Xj,ahighestweightvectoringLetWbethesubspace of§ ‘obtained bysuccessively applying allF's.ThenWisapropersubspace of since itsweight space W,corresponding tofis onedimensional. Bythe argument wehave seen several times, @preserves W.Now ifj=a@4, W would beanideal ing@4q', andthiswould force X,0tobelong to making W,twodimensional again, a Tofinish thisstory, weshould show that thesimple Liealgebras corre: sponding totwodifferent Dynkin diagrams cannot beisomorphic, ie,that thetwochoices made ingoing from asemisimple Liealgebra toDynkin diagram donotchange theanswer. The general facts are: (1)Any twoCartan subalgebras ofasemisimple Liealgebra areconjugate, ie,there isaninner automorphism byanelement inthecorresponding adjoint group, which takes one into theother.@)Anytwodecompositions ofarootsystemintopositiveandnegativerootsdifferbyanelementofthe Wey! group. ‘These arestandard facts which areproved inAppendix D.Both statemeatsaresubsumed inthefactthatany{WoBorelsubalgebras of@semisimple Liealgebra areconjugate, aBorel subalgebra being thesubspace spanned bythe Cartan subalgebra andtherootspaces 9,forpositive «.Forthose readert. ‘whocrave logical completeness butdonotwant togothrough somuch general theory, weobserve thatmost possible coincidences canberuled ott bysuch simple considerations ascomputing dimensions, and others canbé, ruled outbysimple adhoc methods, ef,Exercise 21.17. Finally,wemustalsoprovethe“existence theorem”:thatthereisasimpleLiealgebra foreach Dynikin diagram. Serte’s theorem quoted above givesunifiedproofofexistence. Butwehave seen and studied theLiealgebrasft theclassicalcases(A,)-(D,), anditismoreinkeepingwiththespiritofthes,lectures toatleast trytoseethefiveexceptions explicitly. This isthesubjedofthenextlecture. LECTURE 22 §2and Other Exceptional LieAlgebras Thislectureismainlyaboutgz,withjustenoughdiscussionofthealgebraicconstruc- tionsoftheotherexceptional Liealgebras togivethereaderaSenseoftheircomplexity.92,beingonly14-dimensional, isdifferent: wecanreasonably carryoutinpractice theprocessdescribedin§21.3toarriveatanexplictdescription ofthealgebrabyspecifyingabasis andallpairwise products; wedothisin§22.1 andverify in§22.2 thattheresult really isaLicalgebra. In§22.3 weanalyze therepresentations ofq3,and arrive inpactcularatanotherdescription ofgyittthealgebraofendomorphisms ofatevendimensional vectorspacepreserving generatrilinearform.(Notethat§223maybereadindependently ofeither§22.1,§21.2,or§21.3)Finally,inthefourthsection weWilsketch some ofthemore absiract (Le, coordinate free) approaches tothe construction ofthefive exceptional Liealgebras, While thefst two sections are completely elementary, theconstructions given in§224 involve some fil serious algebra. {22Construction offromitsDynkindiagram{222 Verifying that93isaLiealgebra {223 Representation theoryofos R24:Algebraicconstructions oftheexceptionsLealgebras §22.1.Construction ofg,fromItsDynkin Diagram Inthis section wewillcarry outexplicitly theprocess described inthepreceding ‘ection fortheDynkin diagram (Gq), constructing inthiswayaLiealgebra ‘g2Withdiagram (G.,)(andinparticular proving itsexistence).‘The first step istofind theroot system from theDynkin diagram. Inthecaseofg;thisisimmediate; wemaydrawtherootsystemRc§*associated 340 22.g,andOtherExceptional LieAlgebras tothediagram G,asfollows: 2 OD ay=c4n.tiny|9,=(12.f40) r=adi2) as=02,f8) % 70.9) ms ms ms » me Here thepositive roots aredenoted a,with aanda,thesimple roots. The coordinate system here hasnoparticular significance(inparticular,recallthat theconfiguration ofroots a,andf,isdetermined only upto&realscalar), butisconvenient forcalculating innerproducts.NotethattheWeylgroupisthedihedral group generated byrotation through anangle ofx/3andreflection inthehorizontal; theWeyl chamber associated tothechoice ofordering of theroots given isthecone between theroots aganday ‘Asindicated inthepreceding section, westart byletting X,beanyeigen- vector fortheaction ofhwitheigenvaluea,,andX,anyeigenvectorforthe action ofbwitheigenvalue a.Wesimilarlylet¥,and¥,beeigenvectors witheigenvalues f,and fl,and set Hy=0X,KI)andHy=(X,Y) Wecanchoose¥;and¥,s0thattheelementsH,€satisfyay(H,)=aa(H)= die, (Hy, X,J= 2X, and (Hy, Xg)=2-Xy Ifollows that (iy, Ki)=-2-¥% and (Hy, hy)=-2-%, ie,Hy,Xnand¥;spanasubalgebras,, >sl,C,withH,X;,and¥,anormalizedbasis forthiscopy ofsl,C.Now,itsclearfromthediagramabovethatthereisauniquewayofwritingeach positive roota,asasumofsimple roots a,+°**+a,,0thatthepartial sums a,+"" +a,areroots foreach I<k(modulo exchanging thefirsttwo terms) wegothrough theroot system bythepath 221. Construction ofgfromItsDyokinDiagram u |*| ie, wewrite ay=ay+a3, ayaa tay may tay ba, MgmayHayayHayHay+Oy Ag=ayFaya+0,Ho,Haytay According tothegeneral recipe, thismeans wenow set X= 1% =O Xs), X= (XX Xe=(Xa, Xs), anddefineYy,...,Yesimilarly. TheelementsHy,HayX1500-5XosYasooo»Yo thenform abasis forthe14-dimensional ga,with M7,andff basis forb,X, generator oftheeigenspace 9,,,and¥;agenerator ofgp,forf=I,...,6. The task athand now istowrite down themultiplication table forqin {erms ofthis basis. Ofcourse, some products arealready known: weknow,forexample,thatH,,X;,and¥;formanormalized basisforsly€fori=1,2,{andwehave therelations defining Xs,...,X-and Yj,...,Y,above. In ‘Addition, sinceweknow thattheproduct [X,Xj]liesintherootspace 9,4, foreach iandj,weseeimmediately that[X,,Xj]=0whenevera+ayisnot root. We deduce that (Xi,X5]=1,Xe)=Xa,Xa]=(Xa,X4)=Xa,Xo]=a,X5] =[Xa Xe)=Xa, Xs)=XaXe)=[XsXo)=0, andlikewise Of, %5]=0%, Yo)=Of Ks)=2, 4)=Oe Yo=Oe 5) =(5, Yo)=(a5)=[YaYo)=DissYo]=0. 3a 22.g,andOther Exceptional LieAlgebras Similarly, weknowthat[X,,¥j]=0whenevera,+f=ay~aisnotaroot;this tells usaswell that (4,2)=OX,Yo)=Oa,NI)=XsYe)=On5)=OX,V5) =1%, 4)=1%, 2)=Xs a)=Me, Hh=0 ‘Themultiplication table thusfarlooks like MXMXMH XeXsteXehe i, 0%, 0, + + + * + + + + 6 + 4 sox,88eeeeee xMok0Xe8Xe20200 % On © % # % + 0 0 0% H,0+00X%0O+%+ 0 0 0 0 &+ 0 x+ 4 4 0 0 Oe % re er) Xe + 0 + Oe %+ 0 + 0 x+ 0 8 % + 0 Xs . Thenext thing todoistodescribe theaction ofHyandHH,onthevarious vectors X,and ¥.This canbedone using theinner product onb,butitis perhaps simpler togoback tothebasic idea ofrestriction tothesubalgebras s,,ands,,.Forexample, ifwewant todetermine theaction of7,onthe Various X,,consider howthealgebra 9=bD(a,,@ g,,)decomposes as2 representation ofs,,: ° 3 aug 3 oe 0 8 age 2 oe0ce o (21, Comtacon fom Dyin Diam 16 Mgt wotv epretettion thespan ofXeand Ya sandy noted ‘onecopy oftheadjoint representation Sym?V (thesubalgebra ¢,,itself) spanned byX,,%;,and H,;and twocopiesoftheirreduciblefour-dimensional representationSym*VspannedbyX,,X3,Xe,andX,andY,,Ya,Ys,andYy Inparticular, itfollows that Xz,Xs,X4,and Xyareeigenvectors forthe action ofH,with eigenvalues of—3,—1,1,and3,respectively; andlikewise Ys,YayYs.and Yqareeigenvectors with eigenvalues —3, —1, ,and 3.In similar fashion, weconsider thedecomposition ofqunder the action of %,,=C{Hz, Xz,¥a):diagrammatically, thislooks like 1 1 MtCae a NS“a (,itself),andfourcopiesofthestandardtwo-dimensional representation V,spanned byX,andXs,XyandX,,Y;andYs,andYzandY..Itfollowsthat Foe aa cee intitre din einer andlikewise X5,X,,Ys,and Yqareeigenvectors with eigenvalue —1. Including thisinformation, wecanfillin(hetoptworowsofthemultipli- a “h % Wy =m X% -h Oo er Decomposing g,accordingtotheactionofs,,ands,,givesusinformation about theaction ofX,,X;, ¥%;,and ¥,ontheother basis vectors aswell. Forexample, wesawamoment agothat X,and Xqtogether span asub- Ma 22.gyandOtherExceptional LieAlgebras representation ofgzunder theaction ofs,,,with ad(X,) carrying X,toXg follows from thisthat ad(¥,) must carry Xqback toXq:wehave 2d(¥)(Xe)=24%)24(X3)(X5) =2d(X,) ad(15)(%3) —ad((Xs, DS) =0—ad(H,)(X3) =Xs. Simitarly, since ad(X,) carries X;into —X, which together with Xyspansa ‘copyofthestandardtwo-dimensional representation ofs,,=sly,itfollows that ad(¥,) willcarry —X, back toX,.Likewise from thefact that ad(¥) carries¥;to~Ysweseethatad(¥3)(¥) =~Yi,andsinceadYa):YarYorad(X,): Yor Ys. Wecaninthesame wayusetheaction ofs,,todetermine thevalues ofad(X,)andad(¥2)onvariousbasisvectors,thoughbecausetherepresentationof84,ofg3haslarger-dimensional components this isslightly more com-plicated.Tobeginwith,considertherepresentation of¢4,onthesubspacespannedbyX3,Xs,X4,andX,.Weknowthatad(X,)cartiesX,toX,andsince X;isaneigenvector fortheaction ofthecommutator [X1, %)]=Hy with eigenvalue —3,it follows that ad(¥,) must carry Xyto3Xz: wehave ad(¥,)(%3) =ad(¥,)ad(X4)(%3)=ad(Xy)ad(¥)(%2) ~ad(EX.HDG) =0—ad(HH,)(%s) =3X3. ‘Using this, wecannext determine theaction of¥,onXy ad(¥,)(Xa) =ad(¥%)ad(X4)(%3)=ad(X,)ad(Y)(Xs)—d(H,)(X5) =ad(X,)BXQ) +Xy=4Xy, andwecalculatelikewisethatad(¥,)(X3)=3X4.Analogously,knowingthat ad(¥,)carties¥_10YstoYqtoYsyieldstheinformation thatad(X,)must arty Ys.Ya,and Ys(03%, 4%and, 3%, respectively. Including allthisinformation inthechart,thenextfourrowsofourmultiplication tableare XO NS hh OM WX Mh xX Woy0OX3h4h 03m00i 0%3XoNAX,%3X,000% HW 0 -% 0 0 X& 0 0% % -% 0 0 0 0 O&O ‘Wenexthavetofindthecommutators ofthebasiselementsX,and1,fot 4,§23. Wecannot dothis bylooking attheaction ofthesubalgebra $221,Construction ofg;fromItsDynkinDiagram generated byX,and ¥;,since fori>3wedonotknow thecommutat [%,¥].Rather, thewaytodothisisoutlined inthegeneral proof in( preceding section: wejustusetheexpression oftheX,and¥asbracketsthegenerators Xj,Xz,¥;,and¥;toreducetheproblemtobracketswiththe generators, whichwenowknow.Thus,forexample,thefrstunknownent inthetableatpresentisthebracket[X,Yq].Wecalculatethisbywritingas(X,,Xa],50thatadX)(Y5)=adlEXy,X3D)(%) =ad(X,) ad(X4)(¥5)~ad(X3)ad(X4)(¥%5) =ad(X)(—%) —04(4)8%) =-H, 3H, Likewise, toevaluate [X,, Xu] wehave Ad(XX) =ad(LX, XD) =ad(X,)ad(X3)(X4)~ad(X,)ad(%)(Xa) =—ad(X,)(%s) =—Xe Inthis way, wecan evaluate allbrackets with Xs; knowing these, w canreduce anybracket with X,tooneinvolving X,andXbywritin X4= [X,, Xs}, andsoon.Continving inthisway, wemay complete ot multiplication table: BXORSe ee wR OR mt om IN OG hm ok Oh Oe mo ok my on Oe Oyx reaee x Ce a x a x a a) i Mmm 0 88m % M% -% 0 0 om oO tH, ism 8may 0am x ry 0 Kyo ah ga %ae 5 % wer, 0 sot, ann Ofcourse,inretrospectweseethatthebasiswehavechosenisfarfrom themostsymmetriconepossible:forexample,ifwedividedX,andY,by2 ‘andXs,Xe,Ys,andY,by6,andchangedthesignsofX,andYs,theformol thetablewouldbe M6 22,gyandOtherExceptional LieAlgebras Table221 os Th & S FM Sh Mh 0 mm 4 3% Oh he OM oh Mh Oe mW ay Oh To x moe 0 a Sh wy ty 0 a o -n On om my % - 0 otxn rraarreo" a x ree a ® a a ey % ams,0=O " noe KF x nam 0 ® an) x nm ‘Therewasanothergoodreasonforthesechanges:noweachofthebrackets(%,, ¥]willbethedistinguished element ofhcorresponding totherootayIf wedenote thiselement byH,,then weread offfrom thetable that Hy=Hy43M, Hy2H,+3th, 22) Hy= y+ly, He Hy+2 and HUM [X= 2X, Uh, W= -2%, 23) fort=1,2,3,4,5,6. §22.2. Verifying That g,IsaLieAlgebra ‘Thecalculation oftheprecedingsectiongivesacompletedescription ofwhat theLiealgebra g,must look like, butthere isstillsome work tobedone:unlessweknowthatthereisaLiealgebrawithdiagram(G3),wedonotknowthat theabove multiplication table defines aLiealgebra, letalone asimple ‘one. Infact, thesimplicity isnotmuch ofaproblem (cf.Exercise 14.34), but toknow that itisaLiealgebra requires knowing that theJacobi identityis valid.Onecouldsimplycheckthisfromthetableforall'#)triplesofelements from thebasis, arather uninviting task.‘Thereisanotherway,whichgivesmorestructuretotheprecedingcalcula:tions, and which will give aclue forpossible constructions ofother Li algebras. The root diagram for(G,) ismade upoftwohexagons, onewitJongarrows,theotherwithshort.Thissuggeststhatweshouldfindacopyof thecorresponding LiealgebraslyCinsideg3.Thesubspacespannedby and theroot spaces corresponding tothesixlonger roots isclearly closed ‘under brackets, soistheobvious candidate. Thelongroots areay,,and ig=a5+a3,andtheir inverses. Sowedefine gotobethesubspace spanned bythecorresponding vectors: Go=C{Hs, Hz,Xs,Ys,Xz,Ya,Xo,Yo. AyXs Ys XM % Xe Xe Hs 0 2X5 —2¥%, -X, y Xe —Vs xy rs rnSk A yy Ay 0 Ys % =Xs ° Xe Hy+H, ‘Thisisexactlythemultiplication tableforsl,C,withitsstandard basis(in same order): hy =C{Eyy—Ea.asBa,2—Es,3yBs,20 BatsEasyEs,20Es,a9Baa} ‘Sowehave determined anisomorphism (NoterightawaythatthisverifiestheJacobiidentityfortriplestakenfrom ‘The restoftheLiealgebra must bearepresentation ofthesubalge!{%&#1,andweknowwhatthismustbe:thesmallerhexagon istheuniotthe(wotriangles whicharetheweightdiagrams forthestandard representionofsl,anditsdual,whichwedenoteherebyWandW*;Wisthesum. theroot spaces fora,,f,,and fs,while W*isthesum ofthose forP,. anda. « <PKE »we<J—_ bsaBs Be Again, alook atthetable shows that thevectors X,,¥,,and Y,form aba: forW=C? that corresponds tothestandard basis ¢,,¢;,and es,at 348 22.gzandOtherExceptional LieAlgebras similarly Y.,X,,andXyform abasis forW*=(C2)* thatcorresponds to thedual basis ef,ef,andef:wehave WHC(Xa YeBh WE= Ca,XiXs} 2=90@W@W". With these isomorphisms, thebrackets fox WW and ggxWtWe ‘correspond tothestandard operationsofsly€onC?and(C3)*. NextwelookatbracketsofelementsinW.Notethat[W,W]iscontained inW®,either byweights orbylooking atthetable. Thetable is no & 4% Xe 2%) 2X or eet HY 0 -2% a an) Identifying W=C*,W*=(C2)? asabove,weseethatthebracketWxW-+ W*becomes themap WxWawe=MW, oxwee—2-vAw, Similarly forW*,wehave [W*, W*] <W,andthebracket isidentified with themap Wi WhaWaNW, gxbrrond FinallywemustlookatbracketsofelementsofWwiththoseofW’*,which land ingo.Here thetable is % x Xs X. U+H, 3X 3Kxy Myoma, XYKM 3 Hy, Interms ofthestandard bases, [¢,,ef]=3E,,— Syl. Intrinsically, this mapping CE Wx Weas,C 9) ceanbedescribed bytheformula [v,¢10) =30(w)0 —(ow 24 fore,weWandpeW*. Exercise 22.5%. Show that [¢,@]istheelement ofsly characterized bythe formula $222. Verifying That g,isLieAlgebra Blo, 9],Z)=189(Z-0) forallZ€s1C,whereBistheKillingformongy=sly€.Inotherwords,ifwewrite0:theelement ingy=slysatisfying theidentity Bloe9,Z)=(2-0)forallZeqo=s15C, then thebracket [r, canbewritten intheform [2,9] =18-099. Itisnow arelatively painless task toverify theJacobi identity, since,thanhavingtocheckitfortriplesfromabasis,itsuficestocheckitonofarbitrary elementsofthethreespacesqo,WandW*usingtheabovealgebra descriptions forthebrackets. Wewillwrite outthisexercise, sin same reasoning willbeusedlater.Forexample,forthreeortwoelement: {o,thisamountstothefactthatgo=slyisaLiealgebraandWandrepresentations,ForoneelementZingo,andtwoelementsvandwinW,theJacobiid forthese three elements isequivalent totheidentity Zon w)=(Z-) nwton(Zw), which weknow fortheaction of@Liealgebra onanexterior produc similarly foroneelement ingoand(woinW*. TheJacobi identity forZ€go,0€W,andpeW*amounts to [Z,0*9] =(Z-0)49 +04(Z-9). Applying B(Y,—) toboth sides, and using theidentity B(Y,[Z, ? BUCY, Z},X)this becomes ALY, Z]-0) =(Y(Z-v) +(Z-@Y-oe Since @(LY, Z]-0) =@(¥(Z-0)) ~@(Z (Ye) this reduces to (Z-@)(w)=—9(Z-), for w= ¥-0, which comes from the fact that Wand W* are representations.Fortriples,0,winW,theJacobiidentityissimilarlyreducedtotheide (A Zw) + (0.0w(Z-u)+(wAw)(Z-e)=0 forallz€go,whichamountsto UADA(Z Ww+UA(Ze)Aw(Zu)AvAW =ZWAvAW=0 inNW=C; andsimilarly fortriples from W*. For, we W,and @€W®, noting that Uo,#),0]=-2-[0Ame]=—4-400w)Ao=—4-(0f9w ~(H theJacobi identity forthese elements reads 350 22.9,andOtherExceptional LieAlgebras —4-(@(e)w ~otw)o) =—Lm, @1(0) +Lo,@( (28) Theright-hand sideis —L, @H(0) +Lo,9]()=—Gelo)w —o(w)o) +Bptw)o —o(e)m), which proves thiscase. (This lastlinewastheonly place where weneeded to usethedefinition (22.4) inplace ofthefancier (227))‘ThelastcaseisforoneelementvinWandtwoelements@andyinW". ‘This time identity tobeproved comes down to 4g —oO) =Le,o]-¥ —[v1 Applying both sides toanelement winW,thisbecomes —4-Woho() —POW) =(Lo, HI) —W{Le, @]-). Ifweapply¥/tothepreviouscase(22.8)wehave —4-(oledH(m) —omWE) =—HILw, 0-0) +HALH, @-¥ ‘Andthesearethesame,usingthesymmetry ofthe Killing form: 18-9(C0, ¥)-») =BULe, WsOm,#1)=B(Lw,0](0,VI)=180(E™,0-0) ‘This completes theproof that thealgebra with multiplication table (224) isaLiealgebra. With thehindsight derived from working allthisout,of course, weseethat there isaquicker way toconstruct g,,without any ‘multiplication table: simply start with slC ®W@ W*, anddefine products according totheabove rules. §22.3. Representations ofg) Wewouldnowliketousethestandardprocedure, outlinedinLecture14(andcarried outfortheclassical Liealgebras inLectures 15-20) tosaysomething about therepresentations ofg,.One niceaspect ofthis isthat, working simply from theroot system ofg,andanalyzing itsrepresentations, wewillarriveat ‘What isperhaps thesimplest description ofthealgebra: wewillseethatgisthealgebraofendomorphisms ofaseven-dimensional vectorspacepreserving ageneraltrilinearform.‘Thefirststepistofindtheweightlatticeforg2.ThsisthelatticeAy<5 dualtothelatticeTy<bgeneratedbythesixdistinguished elementsHj.By(222), Tyisgenerated byH,andH,.Since thevalues oftheeigenvalues 1, anda,onH,and1,aregiven by a(Hy)=2, ath)=1, ax(H,)= 3, ay(Ha) =2, itfollows that theweight lattice isgenerated bytheeigenvaluesa,and,(and inparticulartheweightlatticeAyisequaltotherootlatticeA).Thepicture isthus £223.Representations ofg; Asinthecase oftheclassical Liealgebras, theintersection ofthe(clo Weyl chamber #”with theweight lattice isafreesemigroup onthe fundamental weights = 2ay+a, and, =3a,+2a. Anyitteducible representation ofqwillthushaveahighestweightvects‘hich isanon-negative linear combination ofthese two. Asusual, weW T,yfortheirreducible representation with highest weight aco,+barsLetusconsiderfirsttherepresentation I.withhighestweighta.Tr:lating «,around bytheaction oftheWeyl group, weseethat thewei diagram ofFolooks like 392 22.gyandOther Exceptional LieAlgebras Sincethereisonlyonewayofgettingfromtheweight«totheweight0bysubtraction ofsimple positive roots, themultiplicity oftheweight 0inTy. ‘must beI.Ty,9isthus aseven-dimensional representation. Itisthesmaliest oftherepresentations ofgs,andmoreover hastheproperty (aswewillverifybelow)thateveryirreducible representation ofq,appearsinits tensoralgebra;wewillthereforecallitthestandardrepresentation ofg,anddenoteitV.‘The next smallest representation ofgistherepresentation To, with highest weight «,;thisisjusttheadjoint representation, with weight diagram Note that themultiplicity of0asaweightofT,;is2,andthedimensionof To.qis14 ‘Consider next theexterior square A?V ofthestandard representationV=Ty,0ofqs.Itsweightdiagramlookslike from which wemay deduce that AV21. @V. Inparticular, since theadjoint representation I,,ofg2iscontained inAY andtheirreducible representation I,with highest weight aa, +ba, i contained inthetensor product Sym*V@Sym'T'p,,,weseethateveryirreduc iblerepresentation of,appears insome tensor power V®™ ofthestandar: ‘Next,lookatthesymmetric squareSym?Vofthestandardrepresentation a4 22,gyandOther Exceptional LieAlgebras Clearly, this contains acopy oftheirreducible representation T>,0 of with highest weight 2.,. Depending onthemultiplicities ofthisrepresentstion,itmayalsocontainacopyofVitsel,ofthetrivialrepresentation, ofboth; oritmay beirreducible. Toseewhich isin factthecase, weneed to know more about theaction ofq,onthestandard representation v.We willdothisintwo ways, first bydirect calculation, and second using the decomposition ofgsinto sl,@W@W*. Although thesecond approach is shorter, thefist illustrates how onecancalculate fortheexceptional Lie algebras very much aswehave been doing intheclassical eases. Todescribe Vexplicitly, stat with ahighest weight vector for¥,ie,any‘nonzeroelemento,ofthecigenspace V4¢Vfortheactionof)witheigenvalvea.Theimage of0,under therootvector ¥,willthen beanonzeroelement ofthe eigenspace ¥3with eigenvalue a,(this follows from thefactthat the direct sum V@Va,a8&representation ofthesubalgebra 2,,<gsacopyof thestandard representation of, =sl3C), Similarly,theimageofoyunder¥ isageneratoro,oftheeigenspace V,witheigenvaluea,theimageof,under ¥,isagenerator ofthe eigenspace Vgwith eigenvalue 0,and soon.Wemay thus choose asa basis for Vthe vectors % =H = Kl, w= Kl mr AK, y= FOr), and me=—YO), where o,(resp. w)isan eigenvector with eigenvalue o(resp. .).(The signs and coefficient} inthedefinition ofw,arethere forreasons ofsymmetty—seeExercise2.10)Diagrammatically theactionofgymayberepresented bythearrows Moyexe Y, wow eAox eee a woo<—e Exercise 2239. (i)Verify that thevectors 0,w;,and u,asdefined above, ae indeed generators ofthe corresponding eigenspaces (i)Find, interms ofthisbasisfor¥,theimagesofv,undertheelementsYs,Ya,Yo,andYo. Exercise 22.10, Show that theelements X,and ¥;¢g,allcarry basis vectos,1andw,intootherbasisvectors,uptosign(orto2ef0,ofcourse),andcarytotwice basis vectors, that is,Xju=2o,and Yu=20forf=13,4, $223. Representations ofgy as: Now, therepresentation Sym*V has,asbasis, thepairwise products of thebasis veetors forV;andthesubrepresentation T;,isjustthesubspace generated bytheimages ofthehighest weight vector ofunder (repeated applications of)thegenerators F,Y,ofthenegative root spacesof9.Thus, forexample,theeigenspace inSyméVwitheigenvalue a,isthespanoftheproducts u-o, and0°04;thepartofthislyinginTwillbethespanofthe twovectors Ya¥,¥s(02) and¥Ys¥(ob). Wecalculate: ¥%% Yi(e2) =Yo¥s(2oy°04) =Yl203) =404-05 and WYYY(08)=%YQes-e4)=—oye) =204-05 —20-04 Wesee,inother words, that T>,oassumes theweigh awith multiplicity 2,tothatinparticularSym?VdoesnotcontainacopyofV.Similarly, toseewhether ornotSym?V contains acopy ofthetrivial representation, wehave tocalculate themultiplicity oftheweight 0inT>.o. Since anypath intheweight lattice from theeigenvalue 22,to0obtained by fubtracting a,anda,must passthrough ay,wecandothisbyevaluating the products of¥,andY,onthegenerators vjvyanduv, ofthe eigenspace with fipenvalue ay:wehave HH Yelorvs)=—¥,Y(e8)=—%Qu-e,) =420, —202; YH Tureg)=0;WWM(0105)=YYoluees) =—Yu) =2myr0 = YWY (uo4)=YsYalev5+2m04) =(—w-0y +25°04) =204 —uP—Dgva+Dosey; HY YlOy09)=YY;(ues)=Ym;09) ==m0,+Dmy-055 and WY,Yung)=WY(ueoy+2004)=Yold05) =dmyo0y +diy WeeefromthisthattheO-eigenspaceofTisthreedimensional;wethus ‘have thedecomposition Sym?V 21,6 ®C. 356 22.gyandOtherExceptional LieAlgebras Inparticular, wededuce that theaction ofqaonthestandard representation V=C"preserves aquadratic form; andcorrespondingly that thesubalgebra 42€sI(V) =s1,€ isactually contained inthealgebra s0,C. Wewillsethis again inthefollowing section, where wewillgive alternative descriptionsof theexceptional Liealgebras, andagain in§23.3 where wedescribe compact homogeneous spaces forLiegroups. Exercise 22.11. Analyze ingeneral thesymmetric powers Sym'Vofthestan- dard representation Vofgy. Finally, consider theexterior cube\*Vofthestandardrepresentation. The weight diagram is yn. andater weremove onecopy oftherepresentation Towith highest weight2a,(thisisthesumofthethreehighestweightsa4,a,anda,ofV),weaedeft with $223. Representations of 337 This, bywhat wehave seen, can only bethedirect sum ofthestandardrepresentation Vwiththetrivialrepresentation C.Insum,then,weconcludethat AVxT, o@VOC. Note inparticular that, asacorollary, theaction of3onthestandard tepresentation preserves askew-symmetric trilinear form «onV.Itisnot hard towrite down this form: itis alinear combination ofthe five vectors MyAWAOy,0EAWAMasWhAWAD4,0,AUyAWandWyAWyAvg:and thefactthatitispreservedbyX,andX;isenoughtodetermine thecoeficients:wehave Om My NWA DgHO AWA Met M AWA Dy 204AyAWa+2AWyAtee ‘The fact that theaction ofg3onVpreserves theskew-symmetric cubic form ctakes onadditional significance when wemake anaive dimension fount. Thespace /°V ofallsuch alternating forms hasdimension 35,while thealgebra gl(V) ofendomorphisms ofVhasdimension 49;thedifference istxacilythedimension ofthealgebrag,.Infact,wecancheckdirectlythatthelinear map og) +N sendingA€End(V)toA(o)issurjective. Wededucethat«isageneralcubic‘Mternating form [ie, anopen dense subset ofAV corresponds toforms ‘quivalent to. under Aut(V)}, and hence that Proposition 22.12. Thealgebra q,isexactly thealgebra ofendomorphisms of ‘seven-dimensional vector space Vpreserving ageneral skew-symmetric cubic formovon V. 358 22.4,andOther Exceptional LieAlgebras Exercise22.13*.Verifythatthemap@aboveissurjectivebydirectcalculationoftheaction ofgl(V) onwA'V. Exercise 22.14, Asanalternative tothepreceding exercise, analyze skew- symmetric trilinear forms onC*toshow thatforn<7there areonly finitely ‘many such forms, uptotheaction ofGL,C. Verify that theform waboves general inAC" (Infact, there areonly finitely many cubic alternating forms ‘onC*aswell, though thisisfairly complicated; forn>9asimple dimension count shows that there isacontinuously varying family ofsuch forms.) Notethatthecubicformapreservedbytheactionofqsgivesusexplicitlythe inclusion Vvony deduced earlier from their weight diagrams: thisisjustthemap ¥*-»/\'Y aiven bycontraction/wedge product with «,composed with theisomorphism ofVwith V*. Exercise22.15*.Findthealgebraofendomorphisms ofasix-dimensionalvector space preserving ageneralskew-symmetric trilinearform. ‘Wewillseetheform «»again when wedescribe 9,inthefollowing section ‘These calculations using thetable amount tousing alltheinformation thatcanbeextracted from thesubalgebras s,&sl,C ofq3.Using thecopy ofsl,€ that wefound inthesecond section can make some ofthismore transparent. Make theidentification $2=99OWOW? =shCOWOW*. Asarepresentation ofslyC, theseven-dimensional representation Vmust bethesum ofW,W*, and thetrivial representation C.Ifwemake this identification, v=wowrec, itisnothardtoworkouthowtherestofq,acts.Thisisgiveninthefollowing,table: wow ¢ w yoe % x Xw XO Wooo -0nw Wo) ew Weg a) Ay Bee With this identification, wehave w= 1inC,and 224, Algebraic Constructions oftheExceptional LieAlgebras 359 me, MRE Wye eyinW=O mae oe opm etine =(CY Conversely itisnothard toverify that theabove table defines arepresentationofqa,bycheckingthevariouscasesoftheidentity[6nJ-y=€-(1"y)~1(€-))for &,ing,and yinV.Note that thecubic form «»becomes om Se rune? te rere tetretred) ‘This description ofVcanbeused toverify thecalculations made earlier, andalsotostudy ilssymmetric andexterior powers. Forexample, Sym? decomposes over slyC into Sym?W@Sym?W*@Sym’C® WOCOW*@COWOW*=Sym?W@ Sym?W* @C@WOW*@siCOC. ‘Togettheweightsaroundtheoutsidering,theirreducible representation Ts,must include Sym?W, Sym?W*, and sl,C. Checking that Weg, maps Sym?W* nontrivially toW*shows that itmust also include Wand W*.To finish itsuffices tocompute thepart killed byqa,which must fieinthesumofthetwocomponents whicharetrivialforsI,C;checkingthatthisisonedimensional, onerecovers thedecomposition sym'V =1;,0@C. Exercise22.16.Usethismethodtodecompose A\?VandSym?¥. §22.4.Algebraic Constructions ofthe Exceptional LieAlgebras Inthissectionwewillsketchafewoftheabstractapproachestotheconstruc- tionofthe fiveexceptional Liealgebras. The constructions arenotaseasy as youmight wish: although theexceptional Liegroups andtheir Liealgebrashavearemarkable wayofshowing upunexpectedly inmany areas ofmathe- ‘matics andphysics, they donothave such simple descriptions astheclassicalseries.Indeed,theywerenotdiscovered untiltheclassification theoremforced‘mathematicians (oLook forthem. Tobegin with, themethod weused toconstruct gyinthesecond section ‘ofthis lecture canbegeneralized toconstruct other Liealgebras. This istheconstruction ofFreudenthal, whichwedofirst.ItcanbeusedtoconstructtheLiealgebraegforthediagram(Eq).Fromeyitispossibletoconstructe,andand f,.Then wewillpresent (oratleast sketch) several other approachest0theirconstruction. Sinceitisaratherechnicalsubject,probablynotreally x0 2.gyandOther Exceptions LieAlger suited fora firstcours, wewilltouch onseveral approaches rather than ge adetailed discussion ofone The onstruction of, a8sumgo@WeW*thatwefoundinthesecond section works more generally, with very litle change. Suppose foiasem- Simple Liealgebra, and Wisarepresentation ofgg:letW*bethedial representation, andst 8=HOW OW Wealso need maps MW and nsw ofrepresentations ofgo.Weassume these aregiven bytrilinear maps T: Wes Gand 7":MWS G,which means that (Asim) =TH8,9) and He AV)=Te.v9) Weassume these maps arerelated bytheidentity: (AWA 9=olor" —otwo ein forall»,we WandgeW*, Wecanthen define abracket ongbythesamerulesasinthesecondsectionTodescribeit,weletX,,Z,..denotearbitrary elementsofoy8elementsofW,ando,vs9.elementsofW*.The bracket ingidetermined Bysetting: (©LG YI=04.7) _thegivenbracketingo), i)[Xo] =X-e——(theactionofgoonW), Gi)(Xo)=X-@__(thecanonical actionofgyonW*), Gv)[un]=e-(0n w) forascalaratobe determined), ()Ca¥) =B-(@ AW) (forascalartobedetermined) (i)[welse(ere) (forascalaretobe determined),Asbefore,@istheelementofgpsuchthatB049,2)=02-0)forallZeGo, where Bisthe Killing form ongg:The rules ()-vi) determine abiteatproduct[,}onallofg,andthefactthatitisskewfollowsfromthefstthat (XX) =0,(0,«] 0,and(9,9]=0 ‘The argument that wegave showing that satisfies theJacobi identiworksinthisgeneralcasewithoutessentialchange,exceptfortheonecasofatriplewithtwoelementsinWVandoneinW*,wherewehadtodoanexplicit! calculation, Theidentity weneed inthe general case inplace of(2.7) th identity: ab-((w— gmp) =e((ore)-w—(we9)-0, —(2IEh whichisequivalenttotheJacobiidentityforo,wandg.Again,thesplothe resulting Lialgebra iseasy tose,provided alltheweight spaces Onedimensional, using Exercise 14.4, 90wehave: {224AlgebraicConstructions ofthe Exceptional LieAlgebras 3 Proposition 22.19 (Freudenthal). Given arepresentation Wofasemisimy Liealgebra goandtrilinear forms TandT°inducing maps \?W+W*a AW +W,such that(22.17)and(22.18)aresatisfied,theaboveproductsmake 9=%OW@ Wt intoaLiealgebra. Iftheweight spaces ofWareallonedimensional, and11 weights ofW,W*, andtheroots ofgoarealldistinct, andabc#0, then gsemisimple, withthesameCartansubalgebra asgo Exercise 22.20*, (a)Show that thetrilinear map Tdetermines amap / RW W*ofrepresentations ifand only iftsatisfies theidentity T(X-u,6,w)+Tu,Xo,w)+Thws0,X-w)=0 VX€fos andsimilarly forT°. (&)Show that(22.18) isequivalent totheidentity ab-(0.0 Wo A¥)=-(Blwey, 049)—Bowe,v6¥)). ‘TheLiealgebraegfor(Eq)canbeconstructed bythismethod.ThistimBoistakentobetheLiealgebraslg;ifV=€°isthestandardrepresentatiosOfsi,€,letW=A®¥,soW*=A'V*;thetrilinear mapistheusualwedg: product, RV@NVONVINV=C, Andsimilarly forA'V*. Weleave theverifications tothereader: Exercise 22.21*. (i)Verify (22.17), andcheck theconditions ontherootsofst, andtheweightsofAVandA?*.(i)UsethefactthatB(X,¥)=18-Tr(XY {orstytoshow that (22.18) holds precisely if¢~18ab. (ii)Show that the Dynkin diagram oftheresulting Liealgebra is(Eq) Note that thedimension ofslg is80,andthat ofWand W*is84,sothe ‘tumhasdimension 248, aspredicted bytheroot system of(Ey). ‘Once theLiealgebra egisconstructed, e,and e,canbefound assub-‘algebras,a8follows,NotethatremovingoneortwonodesfromthelongarmsiftheDynkindiagramof(B,)leadstotheDyokindiagrams(E,)and(Eg).Ingeneral, ifgis asimple Liealgebra, with Dynkin diagram D,consider a gubdiagram D”ofDobtained byremoving some subset ofnodes, together ithallthelines meeting these nodes.' Then wecanconstruct asemisimple jubalgebra q°ofqwithD°asitsDyokin diagram, Infact, gisthesubalgebra Srverated byalltheroot spaces g4,, whereaisarootinD”, there aredouble ortriplelinesbetween twonodes, bothnodes should beremoved orkept frites: 364 22.q,andOtherExceptional LieAlgebas Exercise 22.27*, Thecenter ofOisRI, which ispreserved byG.Let¥be orthogonal space (oR-1 with respect tothequadratic form N.Then GisimbeddedinthegroupSO(Y)oforthogonal transformations ofY. (a)Definea“crossproduct”xon¥bytheformulavxw=o-w+Blo,w)-lShow that Gcanbeidentified with thegroup oforthogonal (ransforma- tions of¥that preserve thecross product (b)Show that G=Aut(O) actstransitively onthe6-sphere = {Ln Le =, andthesubgroup Kthatfixes |=e,ismapped onto theS-sphere inef ‘bythemapg++g-j. Conclude fromthisthatGis14-dimensional and simply connected. (6)Show that{D€Der(0): D()=0}isisomorphic tostty(4)VerifythattheLiealgebraofderivations ofthecomplex octonians isthe simple Liealgebra oftype (G3). Exercise 22.28*. The octonions can also beconstructed from theCifford algebraofaneight-dimensional vector space withanondegenerate quadraticform.With,S*,andS~asin§203,witho€V,5,€S*,t=9-5eS"chosensothevaluesofthequadraticformsareIoneachofthem asin Exercie 20,50, define aproduct VxV+ ¥,(o, w)i-+ 00 bytheformula vow =(0-t,)-(w-s). Note that ot; €$*,w-s, €S~,$0their product (0°1,)=(W"5,)isbackin¥ (a)Show thatVwiththisproduct isisomorphic tothecomplex octoniaas ©,with unito,,withthemap ++—p(e,)(0) corresponding toconjugationia °. Conversely, starting with thecomplex octonians ©,one can reconstruct thealgebra of§20.3: define A=00O,defineanautomorphism Jof order 3ofAbyJ(x,¥,2)=(2,xy),anddefineaproduct~fromeachsucces, sionoftwofactors tothethird bytheformulas x-y= Xo, y-2=FoR Px= Fok, (b)Show that Aisisomorphic tothealgebra described in§20.3.(©)Identifying s04Cwiththespaceofskewlineartransformations ofO}show thatforeach AinsogC there areunique BandCins04C such that Alcoy)=Bx)oy+x0Cly) forall complex octonionsxandy.Equivalently, ifonedefinesatrilinearfotms (,+Jomtheoctonions by(x, 2)=Tr((x ©y)02)=Trl o(y2 2) (Ax, y,2)+(x,By,2)+(,Cz)=0 forallx,y,2-ShowthatthistrilinearformagreeswiththatdefinedinExerc2049,andthemappingA+-+Bdetermines thetrialityautomorphism’ ofso4Coforder three described inExercise 20.51, B24, Algebraic Constructions oftheExceptional LieAlgebras 365 Exercise 22.29, Define three homomorphisms from therealClifford algebra C=C(O,7)toEndy(0) bysending »€R?=S°Re,tothemaps L,Ry,andTydefinedbyL(x)=¢ox,Ry(x)=x0,andT(x)=v0(x00)=(00x).(a)Show thatthese dodetermine maps oftheClford algebra, andthatthe induced maps SpingR+Co"=C,+Endy() arethetwospinrepresentations andthestandardrepresentation, respectively (b)Verify that T,(x0y)=L(x): L(y) forall0,x,y,andusethistoverifythetriaityformulain(c)oftheprecedingexercise. ‘Thealgebraf,canberealizedasthederivation algebraofthecomplenifica-wonofa27-dimensional JordanalgebraJ.Thiscanbeconstructed astheset‘ofmatricesoftheform aap aby} Bye witha,b,c scalars, anda,Byin©,Theproduct inJisgiven by xoy= Hxy +yx), ‘where theproducts ontheright-hand side aredefined byusual matrix multi- plication. This algebra iscommutative butnotassociative, andsatisfies the entity ((x©x)y)ox =(xox)(yex)Infact,(F,)isthegroup ofauto- ‘aorphisms ofthis27-dimensional space that preserve thescalar product Gy)=Tr(xe y)andthescalar triple product (x,y,2)=Tr((xe y)°2).The Kemel ofthe trace map isanirreducible 26-dimensional cepresentation offa.Fordetailssee[Ch-S],(Tol,[Pos]._Inaddition, there isacubic form “det” onJsuch that thelinear auto- torphisms ofJthat preserve thisform isagroup oftype (Eq). This again shows f,a8asubalgebra ofeg. jETheotherexceptional Liealgebrascanalsobeconstructed asderivationsafappropriate algebras. Werefer forthisto[Ti2}, (Dr], (Fr2}, [Jac2}, andtbereferencesfoundinthesesources.Otherconstructions weregivenbyWit,&[Wa]. Thesimple Liealgebras arealsoconstructed explicitly in[S-K, §1] Seealso [Ch-S}, (Fri), and[Sc]. What little wewill have tosayabout therepresentations ofthefour ‘acxptional Liealgebras besides g,canwait until wehave theWeyl character formula. LECTURE 23 Complex LieGroups; Characters ‘This lecture serves two functions. First and foremost, wemake thetransition back from Liealgebras toLiegroups: in§23.1 weclassify thegroups having asvensemisimpleLiealgebra,andsayWhichrepresentations oftheLiealgebra,asdescribed inthepreceding lectures, ittowhich groups. Secondly, weintroduce in§232 the notionofcharacterinthecontextofLietheory;thisgivesusanotherwayofdescribing therepresentations oftheclassical groups, andalso provides anecessary framework fortheresults ofthe following twolectures. Then in§23:3 wesketch thebeaut interelationships among Dynkin diagrams, compact homogeneous spaces andthe iereducible representations ofaLiegroup. The first two sections areelementary modulo alitle topology needed tocalelate thefundamental groups ofthe casi ‘groups in§23.1. The third section, bycontrast, may appear imposible: itinvolves,at ‘aious points, projective algebraic geometry, holomorphic finebundles, andthet cohomology. Infact,agooddealof§23.3canbeunderstood withoutthesenotions, thereader isencouraged toread a8much ofthe section asseems ineligible. Aal section§23.4givesaverybriefintroduction totherelatedBruhatdecomposition, which isinchudedbecauseofitsubiquityintheiterate. §231: Representationsofcomplexsimplegroups §242: Representation rings andcharacters §23.3: Homogeneous spaces 8234 Brahat decompositions §23.1. Representations ofComplex Simple LieGroups InLecture 21weclassified allsimple Liealgebras over C.This inturn yieldsaclassification ofsimplecomplexLiégroups:awesawinLecture7,franyLiealgebra gthere isaunique simply connected group G,and allother(connected) complexLiegroupswithLiealgebragarequotientsof6by S221.Representations ofComplexSimpleGronps 367 discretesubgroups ofthecenterZ(G).Inthissection,wewillfirstdescribethe sr0ups associated totheclassical Liealgebras, and then proceed todescribe which oftherepresentations oftheclassical algebras wehave described in Part ILlifttowhich ofthegroups. Westart with Proposition 23.1. Foralln>1,theLiegroupsSLCandSpa,areconnected andsimplyconnected. Forn>1,SO,€tsconnected, withmy(80,€) =Z,and¥(80,C) =Z/2forn>3. Proof. The main tool needed from topology isthelong exact homotopy sequence ofafibration. IftheLiegroup Gacts transitively onamanifold M,andHistheisotropy group ofapointPoofM,thenG/H=M,andthe map G>Mbygg:Poisafibrationwithfiber#2.Theresultinglongexact Sequence is,assuming thespaces areconnected, sootn3(0)>my(H)14(G)+(M)(1). (232) (The base points, which areomitted inthisnotation, canbetaken tobethe ‘entity elements ofHandG,andthepoint FainM.)Inpractice wewillknow‘MandHareconnected, fromwhichitfollowsthatGisalsoconnected. Fromthisexact sequence, ifMandHarealsosimply connected, thesame follows for6. Toapplythefongexacthomotopysequenceinourpresentcircumstartoe weargue byinduction, noting first thatSL,€ =SO,€ ={1}.Now consider theaction of G=SL,C onthemanifold M=C'\(0}. The subgroup fixing thevector Py=(1,0, ..,0) consists ofmatrices whose firstcolumn is (1,0,...,0) and whose lower right—1)by(n—1)matrixisinSLy_y(it follows thatastopological spaces H=SL,_,C xC*!. Since Missimplyconnectedforn22(havingthesphereS?*~*asadeformation retract),andHhasSL,-€ asadeformation retract, theclaim forSLy€ follows from (23.2) byinduction onm ‘Thegroup SO,C isisomorphic tothemultiplicative group €*,which has thecircle asadeformation retract, 80x,($0;€) =Z.The group G=SO,C actstransitively onM=(0€C*:Q(0,0)=1},whereQisthesymmetricbi- linear form preserved byG.(The transitivity oftheaction ismore orless {equivalent toknowing that allnondegenerate symmetric bilinear forms are equivalent) For explicit calculations take thestandard Qforwhich the standard basis {¢,)ofC*isanorthonormal basis. This time thesubgroup H fixing e,isSO,_,€. From thefollowing exercise, itfollows that Mhasthe sphere S*-"asadeformation retract. By(23.2) themap 1(80,-,€)~ (50,0) isan isomorphism form24.Soitsulfices tolook atSOsC. This could be done bylookingatthemapsinthesmeexactsequence,butwesawinLecture Othat SO,€ has. two-sheeted covering bySL3C, which issimply connected bythepreceding paragraph, so1,(SO,C) =Z/2,8srequired, 368 23.Complex LieGroups; Characters Thegroup G=Spz4€ acts transitively on M={(0, weC*xC™: Qo, )=1), where@is theskew form preserved byG,and theisotropy group isSp24-2C. Since Sp, =SL,C, thefirstcase isknown. Bythefollowing exercise, since Misdefined inC*byanondegenerate quadratic form, MhasS"! asa deformation retract, soweconclude again byinduction. oO Exercise 23.3%, Show that((¢1,...,7.)€C": Zz?=1)ishomeomorphic to thetangent bundle (othe(n—l)-sphere, ie,16 Toes={(ty2)ESxRYwv=0}. Using theexact sequence {1}+SL,C+GL,C +C* {1}wededuce from theproposition and (23.2) that m(GL,C) =Z. 4 Exercise 235. Show that foralltheabove groups G,thesecond homotopy, groups 3(G) aretrivial Wedigressamomentheretomentionafamousfact.Eachoftheabove‘groups Ghasanassociated compact subgroup: SU(n) ¢SL,C,Sp(7) <SPCandSO(7)<SO,C.Infact,eachofthese subgroups isconnected, andthese inclusions induce isomorphisms oftheir fundamental groups. Exercise23.6.Provetheseassertions byfindingcompatible actionsofthe subgroups onappropriate manifolds, Alternatively, observethatineachcasethecompact subgroup inquestion isjust thesubgroup ofGpreserving a Hermitian form onC"or€*,anduseGram-Schmidt togive aretractionof, Gonto thesubgroup. Now,byProposition 23.1thesimply-connected complexLiegroupscorre-sponding totheLiealgebras g=sI,C, sp,C, ands0,€ are G=SL.C, SpiC, andSpin,C. ‘Wealsoknow thecenter Z(G)ofeachofthese groups. From Lecture 7we. also know theother connected groups with these Liealgebras: +Thecomplex Liegroups with Liealgebra sl,C areSL,C andquotientsof SL,€ bysubgroups oftheform (e2*"-}, formdividing m(inparticular, it's prime theonly such groups areSL,C andPSL,C). +Thecomplex Liegroups with Liealgebra sp,,C areSpay and PSp,,C. +The complex Liegroups with Liealgebra s03,4,C areSpina,Cand SOC. and +Thecomplex Liegroups with Licalgebra s024C areSpin,,C, SO;4C and)PSO,,C;in addition,ifmiseven,therearetwoothergroupscovereddoublybySpin, andcovering doubly PSO,,C [ef.Exercise 20.36]. 231, Representations ofComplex Simple Groups 3 These arecalled theclassical groups Inthe cases where wehave observ coincidences ofLiealgebras, wehavethefollowing isomorphisms ofgrout SpinsC =SL,C andSO,€=PSL,C; SpingC =SL,€ xSL,C and PSO,C =PSL,€ xPSL,C; SpinC =Sp, and SOs€ =PSp,C; and Sping =SL4C and PSO,C =PSL,C. Note that inthefirst case n=4where there isanintermediate subgrov between SL,C andPSL,C, thesubgroup inquestion isinteresting: itturt outtobeSO,C. Ingeneral, however, these intermediate groups seldom aris Consider now representations ofthese classical groups. According toth basic result ofLecture 7,representations ofacomplex Liealgebra gwi correspond exactly torepresentations oftheassociated simply connected Li group G:specifically, foranyrepresentation pa altY) of, setting BlexptX) =exp(oX)) determines awell-defined homomorphism 7:6+GL"). Foranyother group withalgebra g,given asthequotient G/CofGby« subgroupCcZ(G), therepresentations ofGaresimplytherepresentation ofGtrivialonC.Itisthereforeenoughtoseewhichoftherepresentation: oftheclassical Liealgebras described inPart IIIaretrivial onwhich sub groups CcZ(G). ‘This turns outtobevery straightforward Tobegin with, weobserve thatthecenterofeachgroupGwithLiealgebragliesintheimageofthechosenCartan subalgebra <qunder theexponential map. Itwill therefore be ‘enough toknow when exp((X)) =IforX€b;andsince therepresentations ofgareparticularly simpleonbthispresentsnodifficulty.‘Whatwedohavetodofirstistodescribe therestriction ofthe exponential ‘map toh,sothat wecansaywhich elements ofhexponentiate toelementsof “Z(G, Fortheproup thaatgen asmata groupe, twill beprety ‘obvious, butforthespingroupswewillneedtodoalittlecalculation. We{willalsowanttodescribetheCartansubgroupHofeachoftheclassicalgroups G,which istheconnected subgroup whose Liealgebra istheCartan sub- “algebrabof9.ForG=SL,C,Hisjustthediagonal matrices inG,i, H=(diag(z,,...,2,): 2a=I}. Similarly inSp,,€ orSO,,C, H={ding(z),..., 2,.27... 24")), whereas in S05,4C, H=(ding(z,,.°., 29,274... 25",DJ.Imeach ofthese cases the 370 23.Complex LieGroups: Characters ‘exponential mappingfrombtoHisjusttheusualexponentiation ofdiagonal mattices. Tocalculate theexponential mapping forSpin,C, weneed todescribe the elements inSpiny that lieover thediagonal matrices in$O,C. This isnotadifficulttask.Calculating asin§20.2,wefindthatforanynonzerocomplexnumber2andanyI<J-<n,andwithm=2n+1orm=2n,theelements 1 “1 a, (toe inthe Clifford algebra are infact elements ofSpin,C. Moreover, if 1p:SpingC -+SOC isthecovering, theimage p(w(z)) isthediagonal matrix whose jthentry is2,(n+j)th entry is2°?,andother diagonal entries are1. These elements (2) also commute with each other, soforanynonzero complex numbers 23,5% wecandefine WOEaye2)=mag)WaCea)ooMalla) 38) Then p(v(24, 529))=diaglt, 42%,27% ---»2¢2) ifm=2m,while ifm= 2n+1,wegetthesame diagonal matrix butwith a1attheend. Let1=Ex—Exsunan theusual basis forbc $04. Lemma239.Foranycomplexnumbersay,...yyy explay Hy+ +a,H,) =we", ..€%) inSpin,C. PRooF. Since themap exp: }+SpingC isdetermined bythefacts thatitis continuous, ittakes 0to1,and itscomposite with pistheexponential for 80,€, this follows from thepreceding formulas. oO Exercise 23.10°. Show thatexp(5;a)Hj) =ifandonlyifeach ayisin2xiZ andSoa,e4niZ. Weseealso that exp(h) contains thecenter ofSpingC. Indeed 1 w(=1, IB) and ifmiseven, theother central elements are+0,withcw=w(,...,i),aswecalculated inExercise20.36.(This,ofcourse,abocontains thefactthat there isapath between 1and —I,proving again that Spin,,C isconnected.) Exercise 23.11*. Verify forall theclassical groups Gthat: (i)H=exp()isclosedsubgroup ofGthatcontainsthecenterofG;(i)themapoffunda:‘mentalgroupsx,(H,¢)+m,(G,¢)issurjective;i)foranyconnectedcoveringx:G' +G,x-"(H) isconnected andistheCartan subgroup ofG. NowletG=G/Cbeasemisimple LiegroupwithLiealgebragandCarta,subalgebra b.Choose anordering oftheroots, andletTbe theirreduciblerepresentation ofqwithhighestweight2.Thebasicfactthatweneedis $2.1, Representations ofComplex Simple Groups 5 Lemma23.12,Therepresentation [,isarepresentation ofG=G/Cifaonly if AX) 2niZ_ whenever exp(X)e C. Proor. The representation Iisarepresentation ofGwhen g-v=»for+ GEC, where visahighest weight vector inT}.Since exp(6) contains C,th says exp(X)- 0=oforall Xehsuch that exp(X) €C.Now bythenatural ofthe exponential map,andsinceX-»=A(X)oforX€,wehaveexp(X)-» ©,Hencetheconditionisthate"0=v,orthate2)=1ifexp(X)e which isthe displayed criterion, c Letuswork thisoutexplicitly oreach oftheclassical groups. Itmay hel tointroduce anotation forthe irreducible representations which,amongoth Virtues,allowssomecommonterminology inthevariouscases.Notethatfo cachofsl.SP20s$05qand503445therootspaceb*isspannedbyweight wehave called Ly, ...,Ly,80a weight can bewritten uniquely infore AL, +004 Agly. We may sometimes write Ainplace ofthe weigh Auky +--+ Auk. Intherest ofthislecture atleast, wewrite 1forth inreducible representation with highest weight Ly +": +Ayby. Note thabyourchoiceofWeylchambersthehighestweights2=(2y,...,4,) hataris:satisfy AyBay 2 AyZO LOsly $Paus AN039445 wherethe4,areallintegersinthefirsttwocases,andfor$03.45theyareetherallintegers orallhal-integers; and AzazoZASWIZOfors0,,,withthe2,allintegersorallhal-integers. Proposition 23.13. Foreachsubgroup Cofthecenter ofG,therepresentation Tis arepresentation ofG/C precisely under thefollowing conditions: ()G=Slasy€,Chasordermdividingw+1:4,0:modi). Gi)G=Spz,€,C={+1}:DA,iseven. Gi)G=SpingyC orSpingyss€, C=(41): allAareintegers Gv)G=Spin,,€, C=(1,te}: all2,areintegers, 3A,iseven.()G=Spin,,C, neven,C=(1,c}:YA,isaneveninteger;andforC={1,~@}:8:4,—9/2tsanoddinteger. Inparticular, representations ofPSL,»1€ aregiven bypartitions 2with LA,=Omod(n +1),andthose forPSpzy€ have5:4,even.Case(i)verifies ‘that wesawinLecture 19about representations of50,C. RepresentationsofPSO,Ccorrespond tointegralpartitionsAwithJ>4,even. Prot. With thepreceding lemma andtheexplicit description ofeverything‘iasight,thecalculations areroutine.Incase(i),forexample, agenerator for am 23.ComplesLieGroupsCharacters Cis oftheform exp(X), with xeanim(5.Ey)—MEgusa) and soA(X)=(2ni/m)(52,) willbeamultipleof2xiexactlywhenYd,is divisiblebymForSpascexp(X)=—1whenX=mi(S.1)90AX)=x3Ay and (ii)follows. The calculations aresimilar forSpin,C, noting that exp(2ni(if) =—1andexp(ailS H)))=« a Byway ofanexample, reall that anyirreducible representation ofeC i oftheform Sym*V, where Visthestandard two-dimensional representation. Any such representation, ofcourse, lifts tothegroup SL3C; butitfits toPSL,€ =SOsC ifand only ifkiseven (inparticular, the“standard”representation ofSOConC?isthesymmetric squareSym?V).Foranother cxample, wehave seen that anyireducible representation ofap, may be found inatensor product Sym*V@Sym'W,whereVisthestandardfour- dimensional representation ofsp, and Wc A*V thecomplement ofthe (rivial one-dimensional representation. Allsuch representations liftoSp,C, butthey littoPSp,€ =SO,€ ifandonly ifkiseven—equivalenty, ithey arecontained inarepresentation oftheform Sym'W@Sym*(A2W),where Wis the“standard” representation ofSOC. Exercise 23.14, Show thateach ofthese semisimple complex Liegroups Ghas afinite-dimensional faithful representation. ‘The result oftheproposition canbeputinamore formal setting, which brings outafeature that ouralert reader hassurely noticed: thecenter of thesimply-connected formofgisisomorphic tothequotient groupAw/Ayoftheweight lattice modulo theroot lattice. Wenote first that thisabelian group Aw/Ax isfinite. Wehave seen thisfortheclassical Liealgebras. In feneral, wehave Lemma23.15.ThegroupAy/Anisfinite,oforderequaltothedeterminantof the Carton matrix. Proo®.ThesimplerootsaformabasisfortherootlatticeA.Thecorrespond ‘ingelements H,form abasis for T= Z(H 7€R), alattice inb;thisisproved inAppendix D.4, Since Awisdefined tobethe Tati ofelements off that take integral values onTy,thedeterminant et(a(t,) =detiny) isthe index [Aw :Ap]. 0 $241, Representations ofComplex Simple Groupe Inparticular, forthe exceptional groups,Ay/Agistrivialfor(G,),(F), (E4), andcyclic oforder twofor(E) andorder three for(E,). 'nfact, thecenter ofthesimply-connected group isnaturally isomorp! {0thedual ofAy/Ag. Toexpress this, consider thenatural dual ofthisI: ‘group. Thelattice I,defined inthepreceding proofisa sublattice ofthelatti Ty=(Xebea(X)eZ forallaeR). Note that Aywasdefined tobethelattice ofelements of} that take integs values onTy.Itfollows formally from thedefinitions and thefact that Ay// isfinite that wehave aperfect pairing TillgXAw/An+Qf,(X,a)e¥a(X). ‘Theclaim isthatthere isanatural isomorphism from T/T tothecent FG, which isgiven bytheexponential. More precisely, leteg:)-»HG thehomomorphism defined by ee(X) =exp(2RiX), ‘WeclaimthatwhenG=Gisthesimply-connected group,Ker(eg)=Txar¢¢(Tw)is thecenterofG,fromwhichitfollowsthateginducesanisomorphisn T/Tx&ZG), Moregenerally,foranyG=G/C,definealatticeT(G)betweenTyandTyt T(G) =Ker(eg). Thenegdetermines anisomorphism Ty/(G) =Z(G). ‘Wemay thus state our result as ‘Theorem 23.16. There isaone-to-one correspondence between connected Li groups Gwith theLiealgebra gand lattices A<b such that Ape Ay. Thecorrespondence isgiven byassociating toagroup Gthelattice dual toth fernelof theexponential mapexp: ¢-+G;inparticular, thelargest lattice Ay ‘corresponds tothe simply-connected group, thesmallest Agtotheadjoint grow,withnocenter.Intermsofthiscorrespondence, theirreducible representationYofgwithhighest weight 2€h*willlift60arepresentation ofthegroupcorresponding toAcb°ifandonlyifLeA Note also that H=W/T(G)=C8xxCF,Withn=dimchcopiesofC. a 23.Complex LieGroups; Character: Exercise 23.17*. Show that these claims follow formally from what wehave seen: thattheimage oftheexponential map contains thecenter, andthatfor anyweight «there isarepresentation VofgwhoseweightspaceV,isnotzero. Show also that egdetermines anisomorphism I'(G)/T, =*,(G). Indiagram form, ty Gou ‘Center(G) 1 TO). G u (0) 1 Tr é Exercise 23.18. Find thekernels ofeach ofthespin andhalf-spin representa-tionsSpin, -»GL(S)andSpin,+GL{S*) Exercise 23.19*,Classify theirreducible representations ofthefullorthogonal group 0,€. Note that byouranalysis oftheLiealgebra g,there isaunique group G with this Liealgebra, which issimultaneously thesimply-connected andadjointforms;therepresentations ofthisgroupareexactlythoseofthealge- brags.Thesame istruefortheLiealgebras oftype (F,) and(Eg), while (E;) and(E,)cach have (woassociated groups, anadjoint onewith fundamental group 2/2and Z/3, andasimply-connected form with center Z/2andZ/3 respectively Itmay beworth pointing outthat each complex simple Liegroup Gcan berealized asaclosed subgroup defined bypolynomial equations insome general linear group, i-c.,thatGisanaffinealgebraicgroup.Everyirreducible representationG-+GL(V)isalsodefinedbypolynomials inappropriate coordinates. This explains why thewhole subject canbedeveloped from the point ofview ofalgebraic groups, asin[Bor!] and[Hu2]. The Weyl group %,which wedefined asasubgroup ofAut(h*), canbe interpreted interms ofanyconnected Liegroup Gwith Liealgebra g.LetH betheCartan subgroup corresponding to6,andletN(H) bethenormalizer, N(H) =(9€G: gH@"' =H}. Wehave homomorphisms: (LH) +Aut(H)+Aut(5)+Auto"), thefirst defined byconjugation, thesecond bydifferentiation attheidentity, and thethird using theidentification ofband)*viatheKillingform.Fact 14.11canbesharpened totheclaimthatthismapdetermines anisomorphism: NCHYN 590. 2329 ‘WhenGistheadjointformofthe Liealgebra, thisisomorphism isproved inAppendix D.The general case follows, using: $232. Representation Rings andCharacters y Exercise 23.21. Show that ifx:G’-+Gisaconnectedcovering,withCarta subgroups 1H’=x-'(H), then theinduced map N(H'VH’+N(H)/His9 isomorphism. Exercise 23.22, For each oftheclassical groups, and each simple root afin anelement inN(H) thatmaps tothereflection W,inW. §23.2. Representation Rings and Characters Jostaswith finite groups, wecanform therepresentation ring Rofasem simpleLiealgebraorLiegroup:takethefreeabeliangroupontheisomorphist ‘lasses[7]offinite-dimensional representations V,anddividebytherelation (7) =[V'] +[V"] whenever V=V'@ V".Bythecomplete reducibilityrepresentations, itfollowsasbeforethatRisafreeabeliangroupontheclasse(VJofirreducible representations. Again, thetensor product ofrepresenta tions makes Rinto aring: (V]-[W]=[V@W].Manyofourquestion about decomposing representations and tensor products ofrepresentation‘anbenicelyencodedbydescribing Rmorefully.WedothisfirstfortheLialgebras, Forasemisimple Liealgebra g,letA=Aybetheweight lattice, andle Z{A) betheintegral group ringontheabelian group A.Wewrite e(2)forthy basis element ofZ[A] corresponding totheweight 4;fornow atleast thes: arejust formal symbols, having nothing todowith exponentials (but se:(23.40),ElementsofZ[A]areexpressions oftheformY)n,e(J),ie,theyassign Anintegern,toeachweight2,withallbutafinitenumberbeingzero.SoZ[A isanaturalcartierfortheinformation aboutmultiplicities ofrepresentationsDefine acharacter homomorphism Char: R(q)-+ZA] (23.23 bytheformula CharL¥] =F.dim(¥,)e(2), where ¥;istheweight space of fortheweight Aand dim(¥,) itsmultiplicity. This isclearly anadditive homomorphism. Thefistassertion about thischaracter map isthat itisinjective. This comes down tothefact that arepresentation isdetermined bythemultiplicities of itsweight spaces, which issomething wesawinLecture 14, ‘TheproductinthegroupringZ[A]isdeterminedbye(a)-e(B)=e(a+) WeclaimnextthatCharisaringhomomorphism. Thiscomesfromthefamilia fact that VOWn= @YOM, ‘The Weyl group2UactsonZ[AJ,andathirdsimpleclaimisthattheimage ofCharis contained intheringofinvariants Z[A]™. This comes down tothe fectthat, foranirreducible (and hence forany) representation V,theweight 316 23.Complex LieGroups; Characters spaces obtained byreflecting inwalls oftheWeyl chambers allhave thesame dimension Letwy,---,0%beasetoffundamental weights;aswehaveseen,theseare thefirst weights along edges ofaWeyl chamber, and they arefreegenerators forthelattice A.Let Ty, ...,Tybetheclasses inR(g) ofthe irreducible representations with highest weights 7, ..,0 ‘Theorem 23.24 (a)The representation ring R(g) isapolynomial ring onthe variables Fy. Tae (b)The homomorphism R(q)-+Z[A]"isanisomorphism. Inparticular, thissaysthatZ[A]® isapolynomial ringonthevariables Char(T), ...,Char(). Infact, thetheorem isequivalent tothisassertion,sinceifwetakevariablesU,,...,U,andmapthepolynomial ringontheU;toR(g) bysending U,toT;,wehave ZU,-.-Us}+Rig)+ZEA]. Ifthe composite isanisomorphism, thesecond being injective, both must be isomorphisms, which iswhat thetheorem says. Inspite ofits fancy appearance, wewillsethat thetheorem follows quite easily from what weknow about theaction oftheWeyl group 488onthe weightsForanyP¢Z[A}letussaythataisahighestweightforPifthecoefficient ofe(a)inPisnonzero, and, with achosen ordering ofweights asbefore, ais thelargest such weight. Wefirstobserve that ifPisinvariant under 28,then thehighest weight forPisin#” A,where #”isour chosen (closed) Weylchamber. Ingeneral,weightsin#7Aareoftenreferredtoasdominantweights. Now suppose {P;} isanycollection ofelements inZ[AJ®, oneforeach dominant weight 4,such that Phashighest weight Aandthecoefficient of e(A)is 1.Weclaim that theP,form anadditive basis forZ[AJ™ over Z.This iseasy toseeand isthesame argument used inthetheory ofsymmetric polynomials inanyalgebra text: given Pwith highest weight Jifthe coefficientofe(2)ism,thenP—mP,isinvariantwhosehighestweightislower,andonecontinues inductively until onereaches weight zero, ie,theconstants. LetP;=CharT),whichhashighestweighto,,andsupposethecoelficient ofe(a)is1.Sinceanyweight2€9°Acanbeuniquelyexpressedinthe form 2=Jmw, forsome non-negative integers m,andthehighest weightof[]Py"isSmoitfollowsthatthemonomials [](P)"inP,,....P,formanadditivebasisforZ[AJ™.ThissayspreciselythatZ[0,....P,J=ZCAT®, and completes theproof. od Letuswork this outconcretely foreach ofourcases sl,41C, 8P4C, 8030410, and$034. Each lattice Acontains weights wehave called Ly,...,Ly;inthe first case wealso have Lay, with Ly+-" +Lyes =0.Weset §23.2. Representation Rings and Characters a7 x= ely xi"=e(—L,)eZEA] 2325 Note thatincaseLy,...,L,isabasisforA,then ZEA =20 55SeeXTsee MSD EDRs he oeta)A asasubring ofthefield Q(x,, ...,x,): (A,) Forst,,1C, fundamental weights are Ly byt ay bybby+byeeeeyby$+by corresponding totheirreducible representations ¥,A?¥, ....A'V, with V=C**!thestandard representation. Thecharacter of\'Vis¥,e(a), thesumoverallathataresumsofkdifferentL;for1<i<n+1.SoChar(A'V) =Ay,where Ayisthethelementary symmetric function ofxy... Xys1- The Weyl group isthesymmetric group &,,,, acting bypermutation ontheindices, so thetheorem inthiscase says that R(shegs)=ZAI"=ZEAgyoyAud 2326 Note that ZEA] =ZExis sosXaperyMtns Xye1 ~Uy80ZEA] hasan additive basis consisting ofall monomials x*,withaann-tupleofnon-negative integers, butwith notalla,positive. (C,) For sp,C, thelattice Aand fundamental weights have thesame description asinthepreceding case. The corresponding irreducible represen-tationsarethekernelsVofthecontraction mapsMV—>AV,withnowV=C* thestandard representation, k=1,...,mThecharacter ofA*Vis Lela), thesumoverallathataresums ofkdifferent +L,for1<i'<n. The character Char(*V) isthus theelementary symmetric polynomial C,inthe variables x,X;',X35.X3', +5XqyXy!.Thetheoremthensaysthat R(sp29) =ZLAT® =ZC,Cx—W,Cy~ChyooesGyGoad(23.27)=ZC,Cy,Cy... GI (B,) For803,4,€, Aisspanned bytheL;together with (Ly +°-*+ I) ‘The fundamental representations areV,A?V, ...,N'Y, and thespin Tepresentation S.The character ofA'V isthekthelementary symmetric functionofthe2n+1elementsx,x;',...,X—¥,',and1;denotethisbyBy. ‘Thecharacterof5,whichwedenotebyB,isthesumSxf¥+,..-x2"8, where xt =e(Lf2}, x” =e(—L/2). (23.28) ‘SoBisthenthelementary symmetric polynomialinthevariablesx/"+xj". Therefore, (S044€)=ZLAJ™=Z[By,.-»BatsBY. (2329) (D,) For s0,,C, Aand Z[A] arethesame asinthepreceding case. ‘Thefundamental representations areV,A?V, ...,A*-?¥, and thehalf-spin representations S*and S~.The character ofA'V, denoted D,,isthekth dementary symmetric function ofthe2nelements x1,77"... XeXe! The sn 25,Comples LicGroup, Carats characterD*ofS*isthesumY.xf"?-...-x2", wherethenumberofplusigns iseven oroddaccording tothesign. Wehave (60340)=ZLAJ®=©2Dy,---,DaaD*,D-). (23.30) Exercise 23.31*. (a)Prove thefollowing relation inR(s0aqs1€): Ba Bt FB +h, corresponding totheisomorphism S@SzNVO--ONVONY. This describes R602.) asaquadratic extension oftheringZ[By, ....Bi} (b)LetDy(respectively, D;)bethecharacteroftherepresentation whose highest weight istwice thatofD*(resp., D~),s0 that, forexample, thesumof therepresentations Dj}andD;isMY. Prove therelations inR(s0,C): D*-Dt =Ds+Dyg +Dyn to, D>-D- =Dz +Dyy tDy bs DtD>=Dyay +yoy +Duns boo Wecanlikewise describe therepresentation ringforg3.Here, wemaytake asgenerators fortheweight lattice theweights LandL,aspictured inthe diagram andcorrespondingly write Z[A] asZ[xy, xj",x3.x3"Js where x;=e(La Itwillbealittlemoresymmetric tointroduceLy=—L,—Lyaspicturedandxy=x!x5! =e(Ly) andwrite ZLA]=204582,MeerX23—1 $24.2, Representation Rings and Characters wv Inthese terms theWeyl group isthegroup %8generated bythesymmetrt group,permuting thevariablesx,andtheinvolutionsendingeachx,tox;" ‘The standard representation hasweights +L, and 0,and sohascharacter A= AG303) =14Fghaag!bysh. Similarly, theadjoint representation hasweights -£1,, £(Ly~L,) and( (taken twices itscharacter is B= A(XtyX35X3)+ACS[MayX2/X4yXa/X1)- ‘The theorem thus implies inthiscase theequality R(x) =ZEA] =ZEA, B). (23.32 Exercise23.33,VerifydirectlythestatementthatanyelementofZ[x),x3.x3] (.x2%5 —Dinvariant under thegroup2Basdescribedisinfactapolynomia jnAand B. Similarly wecandefine therepresentation ring R(G) ofasemisimple grouyG.WhenGisthesimply-connected formofitsLiealgebraa,R(G)=R(q) $0R(SL,C), R(SP24C), R¢Spin.a4C), and R(Spin;yC) aregiven by(23.26) (23.27, 23.29), and (23.30) Ingeneral, R(G) isasubring ofR(g); wecan reac offwhich subring bylooking atProposition 23.13. Wehave, infact, RGO940)=ZLB.05Bali (2334 R(SOz4C)=Z[Dy,-.+5DyntyDe,Dey (23.35 withDandDzasinExercise23.31,Butthistimethereisonerelation: (DE+Dyan +Daca bo ADE +Dana +Dana bo +1) =(Dyn +Daas to HY Exercise 23.36°. @)Prove 2334) (©)Show that therelation in(23.35) comes from Exercise 23.31(b). Show that R(SO,,€) isthepolynomial ring inthen+1generators shown, modukc theideal generated bytheonepolynomial indicated. (©Describe therepresentation rings fortheother groups with these simple Liealgebras. (@)Prove theisomorphism R(GLC) =Z[E,,..-5 EasEn"), wheretheE,aretheelementary symmetric functionsofx3,+)Xy Exercise 23.37*. (a)Show that theimage ofR(Q,C) inR(SOqC) isthepoly nomial ring Z[B,,..., B,)ifm =2n-+ 1,and Z[D,,.... Dy] itm =2n. 80 23,Complex LieGroups: Characters (b)Show that ROrpo4©)=R(SOsensC1R(Z/2) ZLBosBurBasesBaers ~1) and R(Oz,C) =Z[D,,.--4Dys DanVI where /istheideal generated by(Dz_)? —1andD,Dzx —Dy. Exercise 23.38°. The mapping that takes arepresentation Vtoitsdual ¥* induces aninvolutionoftherepresentation ring:[V]*=[V*].TheringZ[A] hhasaninvolution determined by(e(4))* =e(—2). Show that thecharacter homomorphism commutes with these involutions. Show that forsl...(A))*=Ags;f0F#05441C,and8p25C,and8074€forneven,theinvolutionis theidentity; whilefore0,4€withnodd,(D,)*=D,,(D*)*=D~,(D")*=D* Deduce that allrepresentations ofallsymplectic andorthogonal groups are self-dual. Notethatwhen«istheidentity, allrepresentations areself-dual. Intheother cases, compute theduals ofirreducible representations with given highest weight. ‘The following exercise deals with aspecial property oftherepresentation rings ofsemisimple Liegroups andalgebras. Exercise 2339°. The representation rings R=R(q) and R(G) have another important structure: they areJ-rings. There areoperators A:RG)RG),1=0,1,2,...5 determined by2({V]}) =(A'V] foranyrepresentation V. (a)Show thatthisdetermines well-defined maps, satisfying 2°=1,2"=Id, and Herne Feavoy forany xand yinR.Infact, Riswhat iscalled aspecial -ring: there are formulas for2'(x- y)and4'(/(x)), valid asifxandycould bewritten assums‘ofone-dimensional representations (see,¢.g.,[A-T]}.(b)Show that4‘extends toZ[A],andusethistoverifythatR(G)isaspecial desing, Define Adams operators y*: R-»Rby#*(x) =P,(A'x, ...,4"), where Pris theexpression forthekth power sum (cf.Exercise A.32) interms ofthe ‘elementary symmetric functions, n>k.Equivalently, BE) —WHER) boo=DMA)=0. (©)Show that, regarding Rastheringoffunctions onthegroup G,- (W'x(@) =x(q"). Equivalently, y*(e(4)) =e(ka). (232. Representation Rings andCharacters 28 (6)Show thateach y*isaring homomorphism, andy*0y!=y* {6)Show that forarepresentation V, Char(Sym*¥) =}Char(V) +4W(Char(¥)). Char(\2V) =§Char(¥)? ~$¥2(Char(¥)). Show that Char(Sym*V) andChar(MV) canbewritten aspolynomials in W(Char(V)), Esksd Formal Characters and Actual Characters LetGbeaLiegroupwithLicalgebrag,Foranyrepresentation Vofg,theimage of[¥]€R(g) inZ[A] iscalled theformal character of¥.Asitturns ‘out, this formal character can beidentified with thehonest character ofthe corresponding representation ofthegroup G,restricted totheCartan sub- group H: (2340)1/Char(V) =Y,m,e(a)istheformalcharacter,andexp(X)isanelement OfH,then thetraceofexp(X)onVismae"? This issimply because exp(X) actsontheweight space V,bymultiplication byet, aswehave seen. Inparticular, arepresentation isdetermined bythe character ofitsrestriction toaCartan subgroup, ‘Another common notation for this istosete(X)=exp(2Rix), ande(2)=exp(2niz).Thenthetraceofe(X) isYymge(a(X)) Exercise 23.41. Asafunction onH,thecharacter ofarepresentation is invariant under theWeyl group 20=N(H)/H.DescribeR(G)asaringof SW-invariant functions onH. This isalsocompatible with outdescriptions ofelements ofZ[A]" as Laurent polynomials invariables x,otx}.ForSLys,€, forexample, ifthecharacterChar(W)ofarepresentation WisP(x...»Xsey)thetraceofthetmatrixdiag(2y,..., 240;)0NVisP(2y,..,293)Similarlyfortheothergroups. using thediagonal matrices described inthefirst section ofthis lecture. For thespin groups, theclement w(z),..., 2.) defined in(23.8) hastrace given bysubstituting 2,forx#2,and2)forx," inthecorresponding Laurent polynomial Exercise 23.42*. Ifg,andg,aretwosemisimple Liealgebras, show that RGG1 X93) =R(G1)@ROG) Exercise 23.43*, (a)For thenatural inclusion s1,€<sl,y,C,restrictionof representations gives ahomomorphism R(sl,.,C) +R(Sl,C),whichcanbe a2 23.Complex LieGroups: Characters described bysaying what happens tothepolynomial generators. Since NCOC) =ACHOAC thisis Aim At Ave Give theanalogous descriptions forthefollowing inclusions: Pre2HPC,—$054CC80y—41C, $0341S#0243 ACS HPC, CC HC, SCC 805,C; PCCHAC,—50,44CCHlaqaiC,—$0340Cly,C. (0)The inclusion sk€ xsl,€ ¢sl,,q€ determines arestriction homo-morphism R(sl,,,C)—> R(sl,Cxsl,€)=R(sI,C)@ R(slaC),whichtakespolynomial generators A,(0A,@1+Ay.@Ay+:+1@Ay,Compute analogously for SPCXPaCSPreeely—84CXF0gCC80444C: Which ofthese inclusions correspond toremoving nodes from theDynkin diagrams? Exercise 23.44. Compute theisomorphisms ofrepresentation rings corre: sponding totheisomorphisms s1,C2$05€,s05€=sp4C,andsl,C=504C. §23.3. Homogeneous Spaces Inthissection wewillintroduce and describe thecompact homogeneous spaces associated totheclassical groups. Aswewillsee, these areclassified neatly interms ofDynkin diagrams, andare,inturn, closely related tothe representation theory ofthegroups acting onthem. Unfortunately, weare ‘unable togive here more than thebarest outline ofthis beautiful subject; but ‘wewillaleast try(osay what theprincipal objects are,and what connections among them exist. Inparticular, wegive attheendofthesection adiagram(23.58)depictingtheseobjectsandcorrespondences towhichthereadercantefer while reading this section. ‘Webegin byintroducing thenotion ofBorel subalgebras andBorel sub- groups. Recall first thatachoice ofCartan subalgebra hinasemisimple Lie algebra gdetermines, aswehave seen, adecomposition g=h®DeenGe TO each choice ofordering oftheroot system R=R*UR, Wecan associatea subalgebra b=b0 Qo called aBorel subalgebra. Note that bissolvable, since 9b<Bm BD—Pasegs ete.Infact,bisamaximal solvable subalgebra (Exercise 1435, §233.Homogeneous Spaces 383 IfGisaLiegroup withsemisimple Liealgebra 9,theconnected subgroup BofGwithLiealgebrabicalledaBorelsubgroup. Claim23.48,BisaclosedsubgroupofG,andthequotientG/Biscompact. Proor.Considertheadjointrepresentation ofGong.TheactionoftheBorelsubalgebra bobviously preserves thesubspace b<g,and, infact, bisjust theinverseimageofthesubalgebraofgl(q)preservingthissubspace:ifX=:X, isanyelementofgwithX,€9,andX,¥0forsome aeR,wecouldfindan element Hof <bwith ad(X)(H)¢b—anyHnotintheanniilatorof©h* woulddo,Bisthus(theconnectedcomponentoftheidentityin)theinverse image inGofthesubgroup ofGL(a) carrying binto itself. Itfollows that Bis dosed; and thequotient G/B iscontained inaGrassmannian and hence compact. (Alternatively, wecould consider theaction ofGontheprojective space P(g), andobserve thatBisjusttheconnected component oftheidentityinthestabilizerofthepointcorresponding tothehighestweightspacea,<) Infact, inthecase oftheclassical groups, itiseasy todescribe theBorel subgroups andthecorresponding quotients.ForG=SLysC,Bisthegroupofallupper-triangular matricesinG,ic.,those automorphisms preserving thestandard flag.Itfollows thatG/Bisthe usual (complete) flagmanifold, ic,thevariety ofalllags GB=(Wchemevce) ofsubspaces with dim(¥,) =r ForG=SO;,€ orSOs441€ theorthogonal group ofautomorphisms of 7=C™ orC**" preserving aquadratic form Q,Bisthesubgroup of automorphisms which preserve afixed flag ¥; <-""c¥,ofisotropic sub- spaces with dim(¥,) ~r.Allsuch flags being conjugate, G/B isthevariety of allsuch flags, ie., G/B=(VKeeCC™OY,Ve)=0} NotethatBautomatically preserves theflagoforthogonal subspaces, sothat ‘wecould also characterize G/B asthespace ofcomplete flags equal totheir orthogonal complements, ie, G/B=(0c Vy20°C Vagy©Veg=OM:OWqs)=0}- Finally, thesame holds forSp3,C: theBorel subgroups B&Sp34C arejust thesubgroupspreservingahall-fiagofisotropicsubspaces,orequivalently afulllagofpairwise complementary subspaces; and thequotient G/B is correspondingly thevariety ofallsuch flags. Exercise 23.46. With ourchoice ofbasis {¢,} letV,bethesubspace spanned bythefirst rbasic vectors. IfBisdefined tobethesubgroup that preserves V,for1<r<n, verily that theLiealgebra ofBisspanned bytheCartan subalgebra andthepositive root spaces described inLectures 17and 19. 34 23.Complex LieGroups; Charactes ‘Wenow want toconsider more general quotients ofasemisimplecomplex group G,Tobegin with, wesaythat a(connected') subgroup PofGis parabolic ifthequotient G/P iscompact. Ofcourse, aBorel subgroup Bis parabolic, andsoisanyclosed subgroup containing aBorel subgroup. The following claim isaconverse tothis: itasserts that theBorel subgroups are ‘exactly theminimal parabolic subgroups. (Claim 23.47. IfBis aBorel subgroup andPaparabolic subgroup ofG,then there isanxe Gwith BoxPx! Wewillnotprove thishere, butwilremark that itisclosely related to Claim23.48.IfP<Gisaparabolicsubgroup,thenthequotientG/Pmaybe realized asanorbit oftheaction ofGonPVforsome representation VofG (inparticular, G/Ptsaprojective algebraic variety). Thefistclaim follows from thesecond byaversion ofBore’s fixed point theorem: ifBisaconnected solvable group, Varepresentation ofBandXcPVaprojective varietycarriedintoitselfundertheactionofBonPY, then Bmust have afixed point onX.This isstraightforward: weobserve(by Lie's theorem (9.11)) that theaction ofthesolvable group BonVmust preserveaflagofsubspaces Ochewehav with dim(¥) =i.Wecanthus findasubspace F<Vfixed byBsuch thatX intersects PY,inafinite collection ofpoints, which must then befixed points fortheaction ofBon X.Intheother direction, wewill soon seedirectly howG/Pisaprojective varietywheneverPisasubgroupcontaining B.‘Wecannowcompletelyclassifytheparabolicsubgroupsofasimplegroup, uptoconjugacy. Bytheabove, wemay assume that Pcontains aBorel ‘subgcoup B.Correspondingly, itsLiealgebra pisasubspace ofgcontainingbandinvariantundertheactionofBong;i.e,itisadirectsum P-0Da forsome subset TofRthat contains allpostive roots. Now, inorder forpobeasubalgebra ofg,thesubsetTmustbeclosedunderaddition(thatisiftwo roots areinT,then either their sum isinToFisnot aroot). Since, in addition, Tcontains allthepositive roots, wemay observe that ifa,B,and7 arepositive roots with a= +y,then wemust have -aeT = —feTand—yeT. "WisgenerafactthatPmustbeconnectedifGPIcompact. £233. Homogeneous Spaces 28s Clearly, any such subset Tmust begenerated byR*together with the Negatives ofasubset£ofthesetofsimpleroots.Thus,ifforeachsubsetE ofthesetofsimple roots weletT(Z) consist ofallroots which canbewritten assumsofnegatives oftherootsinE,together withallpositive roots,andform thesubalgebra PZ) =b® Ber (23.49)"E) Py (23.49) then p(Z) isaparabolic subalgebra, thecorresponding Liegroup P(E) isa parabolic subgroup containing B,and weobtain inthisway alltheparabolic subgroups ofG.Wecanexpressthisastheobservationthat,uptoconjugacy, parabolic subgroups ofthesimple group Gareinone-to-one correspondencewithsubsetsofthenodesoftheDynkindiagram, ie.,withsubsetsofthesetof simple roots. Examples. Inthecase ofsl,C, there isasymmetry intheDynkin diagram, so thatthere isonlyoneparabolic subgroup other than theBorel, corresponding tothediagram o——e This,inturn,givesthesubsetoftherootsystem corresponding tothesubgroup 00 « andthehomogeneous space G/P =P? Inthecaseofsp,€, there aretwosubdiagrams oftheDynkin diagram: ep w ante thesecorrespond tothesubsetsoftherootsystem 386 23.ComplexLieGroups;Character co e\;\e (Here weareusingablackdottoindicateanomittedsimpleroot,awhitedot toindicateanincludedone.)Thecorresponding subgroups ofSp,Carethosepreserving thevector e,,andpreserving thesubspace spanned byeyatide3, respectively. The quotients G/B arethus thevariety ofone-dimensional iso- tropic subspaces (ie,thevariety P*ofalltheone-dimensional spaces) and thevariety oftwo-dimensional isotropic subspaces. Exercise 23.50. Interpret thediagrams above asgiving risetoparabolicsubgroups ofthegroupSO,€ofautomorphisms ofC?preservingasymmetric bilinear form. Show that thecorresponding homogeneous spaces arethevatietyofisotropicplanesandlinesinC°,respectively. Inparticular, deducetheclassical algebraic geometry facts that: {)The variety ofisotropic 2-planes foranondegenerate skew-symmetric bilinear form onC*isisomorphic toaquadric hypersurface inP4.(ii)Thevarietyofisotropic2-planesforanondegenerate symmetric bilinearform onC?(equivalently, linesonasmooth quadric hypersurface inP*) isisomorphic toP?. Ingeneral,itisnothardtoseethatanyparabolicsubgroupPinaclassicalgroup Gmay bedescribed asthesubgroup that preserves apartial flagin thestandard representation. Inparticular, amaximal parabolic subgroup,corresponding toomittingonenodeoftheDynkindiagram,maybedescribed asthesubgroup ofGpreserving asingle subspace. Thus, forG=SLC, the kthnode oftheDynkin diagram o—0—*—0 00000 corresponds totheGrassmannian G(k,m)ofk-dimensional subspaces ofC*,(Note that thesymmetry ofthediagram reflects theisomorphism ofthe Grassmannians G(k, m)andG(m —k,m))ForSpa,C,thekthnodeoftheDynkindiagram 000000 0oxi £233. Homogeneous Spacer 87 corresponds tothe Lagrangian Grassmannian ofisotropic kplan, fork =1, 2, ...,n.Similarly, forG=SO,,,,€, thekthnode oftheDynkin diagram ‘corresponds totheorthogonal Grassmannian ofisotropic k-planes inC?**".Finally,forSO,,C,fork =1,2,...,n—2thekthnodeoftheDynkindiagram 0-08-0000 yieldstheorthogonal Grassmannian ofisotropic k-planes inC?*,butthereis. ‘oneanomaly: either ofthelasttwo nodes gives oneofthetwoconnected components oftheGrassmannian ofisotropic meplanes Exercise 23.51*.Compute p(£)directly foreachoftheclassical groups, andverify theabove statements. Why istheorthogonal Grassmannianofisotropic (a— 1}-planes inC2"notincluded onthelist? Aswesaw already inExercise 23,50, thelow-dimensional coincidences between Dynkin diagram ean beused torecover some lacs wehave seen before. For example, thecoincidence (D,) =(A,) x(Ay) identifies thetwo family oflines onaquadratic surface inP®with two copies ofP!.The ‘coincidence (A,) =(D3) war. fives risetotwoidentifications ofmarked diagrams: wehave wd corresponding tothe isomorphism between theGrassmann varieties P= G(1,4),P=G(3,4)andthetwocomponents ofthefamily of2-planes quadric hypersurface Qitself. Finally, anobservation that isnotquite so says thateither connected component ofthevariety of3-planes onasmooth‘quadrichypersurface QinP”isisomorphic [email protected] isanother waytorealize thecompact homogeneous spaces associated toasimple group G.Let V=[),beanirreducible representation ofGwith highest weight A,andconsider theaction ofGontheprojective spacePY:Let pePYbethepoint corresponding totheeigenspace with eigenvalue 4We Claim23.52.TheorbitGpistheuniqueclosedorbitoftheactionofGonPY. Proor.ThepointpisfixedundertheBorelsubgroup B,sothatthestabilizerofpisaparabolic subgroup P,;theorbit G/P, isthus compact and hence. closed. Conversely, bytheBorel fixed point theorem, anyclosed orbit ofG contains afixedpointfortheactionofB;butpistheuniquepointinPVfixed byB. is} Infact,itisnothardtosaywhichparabolicsubgroup P,is,intermsoftheclassification above: itistheparabolic subgroup corresponding tothesubset of= simplerootsthatareperpendicular totheweightA.Now,setsEofsimple rootscorrespond tofaces oftheWeyl chamber, namely, thefacethatisthe, intersection ofallhyperplanes perpendicular toallrootsin£. 7Wethus have acorrespondence between faces oftheWeyl chamber and parabolic subgroups P,such that ifV=I,istheirreducible representation $233. Homogeneous Spaces x isoftheform G/P, where Pistheparabolic subgroup corresponding toth open faceof¥containing 2.Inparticular, weights intheinterior oftheWet ‘chamber correspond toP,=B,and sodetermine thefullag manifold G/l whereas weights ontheedges give rise tothequotients ofGbymaxim: parabolies. Note that wedoobtain inthis way allcompact homogeneou spaces forG. Forexample, wehave therepresentations ofSLyC: aswehave seen, threpresentations Sym*VandSym'V*,withhighestweightsontheboundatieoftheWeylchamber,haveclosedorbits{o*},4yand{I}r-yisomorphic tPYandPV®.Bycontrast,theadjointrepresentation—the complement 0thetrivial representation inHom(V, ¥)=V@V*—has asclosed orbit thy variety oftraceless rank 1homomorphisms, which isisomorphic tothefla,manifoldviathemapsendingahomomorphism @tothepair(Img,Ker@)‘Thepicture is ~, seemmeon Sya'y* Tare cnet P? | etog AR saewnveerwvers= rh . —y a repens $ymyTarecredo Ingeneral,ifVisthestandardrepresentation ofSL,C. intherepresenta- lions ofSL,€ oftheform W=Sym*V wesaw that thevectors oftheform {0*}-ey formedaclosedorbitinPW,calledtheVeroneseembeddingofP*"' Likewise,inrepresentations ofthe form W=A*V thedecomposable vectors 0,A0,0+" 9%}formed aclosed orbit inPW; thisisthe Plicker embedd- ingoftheGrassmannian. Similarly, wemay identify theclosed orbits inrepresentations ofSpaC. Recall here that thebasic representations ofSpyC arethestandard represen- tation V=C*and thecomplement Wofthetrivial representation inthe exterior square (?V; allother representations arecontained inatensor product ofsymmetric powers ofthese. Now, SpaC actstransitively onPV: theclosed orbit isallofP*.Ingeneral, inP(Sym'V) theclosed orbit isjust thesetofvectors(0"}.,» P?.Bycontrast,theclosedorbitinPWisjusttheintersection ofthehyperplane PW<P(/\?V)withthelocusofdecomposable vectors {0AW},.wevi thisisthevariety 390 23.ComplexLieGroups;Characters X= (0.0 w:(0,w) =O} ofisotropic 2-planes A<Vfortheskew form Q. eAclodesquneTreo nPW . («ee = ‘epson Syl“vecsearP Forthegroup Spinyas€; theclosed orbit ofthespin representation Sistheorthogonal Grassmannian ofn-dimensional isotropicsubspacesofC™"", Thecorresponding subvariety G/P =P(S) isavarietyofdimension(n +1)n/2inP™,N=2*—I,calledthespinorvariety,orthevarietyofpurespinors.SimilarlyforSping,C,thetwospinrepresenta-tionsS*andS~giveembeddings ofthetwocomponents oftheorthogonalGrassmannian ofn-dimensional isotropic subspaces of™,oneinP(S* one inP(S"). These spinor varieties have dimension n(n—1)/2.in projectivespaces ofdimension 2"-" —1 Exercise 23.53. Show that thespinor variety forSpin,,_€ isisomorphic each ofthe spinor varieties forSpingsC. Infact they areprojectively equivalent assubvarieties ofprojective space PX,N= 2°" —1. Itfollows that, form<8,thespinor varieties forSping areisomorphictohomogeneous spaceswehavedescribedbyothermeans.Thefirstnewoneisthe10-dimensional variety inP'*,which comes from Spiny orSpin,C- Itisworthgoingbacktointerpretsomeofthe“geometric plethysm”of earlier lectures (eg,Exercises 11.36 and 13.24) inthis light. Finally, wecandescribe (atleast oneof)thecompact homogeneous space forthegroup G;inthisway. Tobegin with, Gyhastwo maximal parabolie subgroups, corresponding tothediagrams oe te E|D 9233, Homogeneous Spnces 21 ThesearethegroupswhoseLiealgebrasaretheparabolicsubalgebras spannedbytheCartan subalgebra b<gtogether with theroot spaces corresponding tothe roots inthediagrams and Inparticular, each ofthese parabolic subgroups willhave dimension 9,so thatboth thecorresponding homogeneous spaces willbefive-dimensional ‘atieties. Wecanusethistoidentify oneofthese spaces: ifVisthestandard seven-dimensional representation ofG,,theclosed orbit in PV=P®willbe ‘hypersurface, which (since itishomogeneous) canonly beaquadric hyper- surface. Thus, thehomogeneous space forG,corresponding tothediagram ——s isaquadrichypersurface inPS.Inparticular, weseeagainthattheactionofG,0nVpreserves anondegenerate bilinear form, ie,wehave aninclusion 6,&+80,, Theotherhomogeneous space¥ofgislessreadilydescribed.Oneway todescribe itistousethefact that theadjoint representation Wofg3is 392 23.Complex LieGroups: Characters contained intheexterior square \*Vofthe standard. Since theGrassmannianG(1,2)<P(MY)oflinesinPVisclosedandinvariantinP(A?V),itfollowsthatYiscontained intheintersection ofGwiththesubspace PIV<P(/\?V)Inotherwords,intermsoftheskew-symmetric trilinearformwonVpreservedbytheaction ofG,,wecansaythat¥iscontainedinthelocus, Ea (Ac Via(A,A,) =0}<G02, Problem 2354.18 ¥=E? Exercise 23.55. Show that therepresentation ofEywhose highest weight i thefirst fundamental weight o»,determines a16-dimensional homogeneous space inP?, ‘Thesehomogeneous spaceshaveanamazingwayofshowingupasextremalexamples ofsubvarieties ofprojective spaces, starting with adiscovery of Severi thattheVeronese surface inP?istheonly surface inP*(nonsingular and not contained inahyperplane)whosechordsdonotfilupP*,Forrecent ‘workalongtheselines,see[L-VdV],withitsappendixbyZakoninterestingprojective varieties that arise from representation theory. Although wehave described homogeneous spaces only forsemisimple Lie groups, this isnoreal loss ofgenerality: anyirreducible representation Vofa Liegroup Gcomes from arepresentation ofitssemisimple quotient, upto ‘multiplying byacharacter (see Proposition 9.17),and thischaracter does not change theorbits inP(V). Itispossible t0take this whole correspondence one step further and useittogive aconstruction oftheirreducible representations ofG;thisis themodern approach toconstructing theirreducible representations, dueprimarilytoBorel,Weil,Bott,and,inamoregeneralsetting,Schmid.Wedorot have themeans todothis indetail inthepresent circumstances, butwe will sketch theconstruction. The idea isvery straightforward Wehave just seen that forevery irreduc-iblerepresentation VofGthereisauniqueclosedorbitX=G/Pofthe action ofGonPY.Weobtain inthiswayfrom Vaprojective variety Xtogether withalinebundleLonXinvariantundertheactionofG(therestriction oftheuniversal bundle from PV) Infact, wemay recover Vfrom thisdata simplyasthevectorspaceofholomorphic sectionsofthelinebundleLonX.Whattiesthisalltogether isthe factthatthisgives usaone-to-one correspondence, between irreducible representations ofGandample (positive) linebundleson, compact homogeneous spaces G/P. More generally, using theprojection maps G/B -»G/P, wemay pullback allthese linebundles toinebundles onG/B,Thisthenextendstogiveanisomorphism betweentheweightlaticeofgand,thegroup ofline bundles onG/B, with thewonderful property thatfot dominant weights 2,thespace ofholomorphic sections oftheassociated line, bundle L,isthe irreducible representation ofGwith highest weight 2 £233. Homogeneous Spaces a Thepoint ofallthis, apart {rom itsintrinsic beauty, isthat wecangbackward:startingwithjustthegroupG,wecanconstructthehomogeneou‘space G/B, and then realize alltheirreducible representations ofGasco homology groupsoflinebundles onG/B.Tocarrythisout,startwithaweighAeb*forg.Wehave seen that4exponentiates toahomomorphism H+C* i.e,itgivesaone-dimensional representation C,ofH.Wewanttoinducethi representation {rom Hto G.IfH¢BcGisaBorelsubgroup,therepresenta tion extends trivially toB,since Bisasemiditect product ofHand thi nilpotent subgroup Nwhose Liealgebra isthedirect sum ofthose g,foi positive roots a.Then wecanform Ly=6 x90 =(GxC)A(9, 0)~(x,x0),x€B), which, with itsnatural projection toG/B, isaholomorphic linebundle ontheprojective varietyG/B.Thecohomology groupsofsuch aline bundle are finite dimensional, and since GactsonL.,,thesecohomology groupsarerepresenta- tions ofG. ‘Wehave Bott's theorem forthevanishing ofthecohomology ofthisline bundle: Claim 23,56. H'(G/B, L,)=0fori#12), where i(A)isaninteger depending onwhich Weyl chamber Abelongs to.IfAis adominant weight(ie.,belongs totheclosureofthe positive Weyl chamber forthechoice ofpositive roots usedindefining B),theni(—4)=0.Inthiscase thesections H°(G/B, L_.,) areafinite-dimensional vector space, onwhich G aos Claim23:57.ForLadominantweight,thespaceofsectionsH°(G/B,L..)istheirreducible representation with highest weight 2 Inthiscontext theRiemann—Roch theorem can beapplied togive afor- ‘mulaforthedimensionoftheirreduciblerepresentation. Infact,thedimension partofWeyl’scharacterformulacanbeprovedthisway.Morerefinedanaly- fis,usingtheWoodsHolefixedpointtheorem,canbeusedtogetthefullchar-cacterformula(cf.[A-B]).Foraveryreadable introduction tothis,see[Bot]‘Weconclude thisdiscussion bygivingadiagramshowingtherelationships‘amongthevariousobjectsassociatedtoanirreducible representation ofasemi-‘simple Liealgebra g.The objects andmaps indiagram (23.58) areexplained next. First ofall, aswehave indicated, the term “Grassmannians” means the ordinary Grassmannians inthecase ofthegroups SLC, andtheLagrangian ‘Grassmannians andtheorthogonal Grassmannians ofisotropic subspaces in ‘thecases ofSp,,C andSO,,€, respectively Likewise, “flag manifolds” refers‘fothespacesparametcizing nestedsequences ofsuchsubspaces. Inthecases. 3 fF geot X\ g » gog -4en ae 2. 3 a8{23-7 sg foes 3 £234. Bruhat Decompositions 395 oftheexceptional Liealgebras,theterm“Grassmannian” shouldjustbeignored;exceptforthequotientofG,byoneofitstwomaximalparabolicsub- ‘groups, thehomogeneous spaces forthe exceptional groups arenotvarieties With which wearelikely tobeapriori familar. With this said, wemay describe themaps 4,B,etc, asfollows: A,A':the map Aassociates toasubset ofthenodes oftheDynkin diagram (equivalently, asubset Softhesetofsimple roots) theface oftheWeyl chamber described by 0,2)>0,¥aeS; ws{ulnarovest, where(,istheKillingform:theinverseisclear. B,B:themap Bassociates toaface#5oftheWeyl chamber thesubalgebra ‘spanned bytheCartan subalgebra b,thepositive rootspaces g,,aRslandtherootspacesg.corresponding tothosepositiveroots«perpendic- ular toWz. Equivalently, interms ofthecorresponding subset Softhe simple roots, gswillbegenerated bytheBorel subalgebra, together with theroot spaces g..foraS.Again, since every parabolic subalgebra is conjugate tooneofthisform, theinverse map isclear. GC: The map Csimply associates toaparabolic subalgebra p<9 the‘quotientG/PofGbythecorresponding parabolicsubgroupP<G.Intheother direction, given thehomogeneous space X=G/P, with theactionof G,thegroup Psustthestabilizer ofapoint inX.Note thattheconnected component oftheidentity intheautomorphism group ofG/P may be strictly larger: forexample, P?*-' isacompact homogencous space for ‘SpaxC, andwehave seen thataquadric hypersurface inP*isahomoge- neous space forGy. D,D:The map Dassociates totheirreducible representation Vofqwith highest weight 2theopen face oftheWeyl chamber containing 4In theother direction, given anopen face 5of¥,choose alattice point AeWyOAy andtake V=Ty. E:Wesend therepresentation Vtothesubalgebra orsubgroup fixing the highest weight vector ve¥. F,F:Weassociate totherepresentation Vthe(unique) closed orbit ofthe corresponding action ofthegroupGontheprojectivespacePV.Goingin theother direction, wehave tochoose anample linebundle Lon thespace G/P, andthen take itsvector space ofholomorphic sections. 23.4. Bruhat Decompositions Wend thislecture with abrief introduction totheBruhat decomposition of semisimple complex Liegroup G,andtherelated Bruhat cellsintheflag Ianifold G/B. These ideas atenotused inthiscourse, butthey appear sooften thewhere that itmay beuseful todescribe them inthelanguage wehave 396 23,ComplexLieGroups;Charactes developed inthislecture. Wewillgive thegeneral statements, butverify themonlyfortheclassicalgroups.Generalproofscanbefoundin[Bor]or{Hv2}. [Aswehaveseen,achoiceofpositiverootsdetermines aBorelsubgroupB andCartan subgroup 1,with normalizer N(H), soN(H)/H isidentified with theWeyl group 28.Foreach We%8fixarepresentative myinN(H),The double coset B-nyB isclearly independent ofchoice ofny,and willbe denoted BW:B. ‘Theorem23.59(BruhatDecomposition). ThegroupGisadisjointunionofthe128) double cosetsB-W:B,asWvarlesovertheWeylgroup. Letusfirst seethisexplicitly for G=SLC. Here N(H) consists ofall ‘monomial matrices inSL,C, ie,matrices with exactly one nonzero entry in each row and each column, and 28=qj amonomial matrix with nonzeroentryintheo(j)throwofthejthcolumnmapstothepermutation ¢.Toseethatthedouble cosets cover G,giveng€G,useelementaryrowoperationsby leftmultiplication byelementsinBtogetanelementb-g°*,withb€Bchosen sothatthetotal number ofzeros appearing atthe lftintherows inbg" is aslargeaspossible.tworowsofb-g"?hadthesamenumberofzsatthe left,onecould increase thetotal byanelementary rowoperation.Sincealthe rows of bg"start with different numbers ofzeros, this matrix canbe putinupper-triangular form byleftmultiplication byamonomial matri, therefore, there isapermutation @sothatb’=n,b-"" isupper triangular, ie, 9=WY -n,:b isinB-o-B. Toseethat thedouble cosets aredisjoint, suppose my=H-n,-b forsome band 6’inB.From theequation b= (1) (WY nyonesees thatbmusthavenonzeroentriesineachplacewhere (nny"*-ny: does, from which itfollows that o”=«.Tnfact,thiscanbestrengthened asfollows.LetU(resp.U~)bethesubgroup ofGwhose Liealgebra isthesum ofallroot spaces 9,forallpositive (resp negative) rootsa. ForG=SLC,U(resp.U)consistsofupper-(resp.lower), triangular matrices with 1’sonthediagonal. For Win theWeyl group, define subgroups U(W)=Ucrn:Ustiy!,UW=UngUntofU,whichareagainindependent ofthe choice ofrepresentative myforW: Corollary 23.60.EveryelementinB-W-Bcanbewrittenuty:bforwieelementsuinU(W)andbinB. Toseetheexistence ofsuch anexpression, note first that theLiealgebmy ofU(W) isthesumofallrootspaces a,forwhich «ispositive andW~"(aht negative; and theLiealgebra ofU(WY isthesum ofallroot spaces 9,fot,whichaandW~!(a)arepositive.OneseesfromthisthatU(W):U(WYHittheentireBorelgroupB.SinceHny=ye:andU(W)'-My =tyU,and Hand Uaresubgroups ofB, £34. Beohat Decompositions 307 Bny:B= UW) UW) HotyB=UW)UW)tyB =UW) ny B. ‘Toseetheuniqueness, supposethatty=w-MybforsomeuinU(W)andbinB.Thenrig!-u-nyisinU~0B={1},80w=1,a8requied. Note inparticular that thedimension ofU(W) isthecardinality ofR*oy W(R"), where R*andR”arethepositive andnegative roots; thisisalso therninimumnumberI(W)ofreflections insimplerootswhoseproductisW,cf.Exercise D.30. Itisageneral fact, which wewillseefortheclassical groups, tatU(W) isisomorphic toanaffine space C". Itfollows from theBruhat decomposition that G/B isdisjoint union of thecosets Xy=Bmp”B/B,again with Wvarying over theWeyl group. TheseXvatecalledBruhatces.FromthecorollatyweseethatXyisisomorphictothe affine space U(W) =C%™,»ForG=SLCandin18~Sy,thegroupU(o)consistsofmatriceswithI'sonthediagonal, andzero entry inthei,jplace whenever either i>jor€°'G)<0""()),whichisanaffinespaceofdimensionIfo)=#{(,j):1>jand ali)<o(/)}- Exercise 23.61. Identifying SL.4C/B with thespace ofallflags, show that X, consists ofthose fags 0.<¥,©Vc: such that thedimensions ofinter~ sections with thestandard flagaregoverned byo,inthefollowing sense: for each I<k<m, thesetofknumbersdsuchthat¥,nC" SKC! is ‘preciselytheset{o(1),0(2),..,o(K)}- Wewill verity theBruhat decomposition forSp,,C byregarding itasa subgroup ofSL,,€ andusing what wehave justseen forSLC, following [Ste2}. Our description ofSp,,C inLecture 16amounts tosaying that it Wsthefixed point setoftheautomorphism @ofSL,4C given by@(4)=ACM, withM=(°{TeBorelsubgroupofSp,€willbethe baersectionofthe Borel subgroupBofSL,CwithSp,C,providedwechange teorder ofthebasis ofC710€15 sy Cn +»ann» 80that Bconsists of [Ratrices whose upper leftblock isupper triangular, whose lower lftblock is 40, and whose lower right block islower triangular. The automorphism @ ‘maps thisBtoitself, andalsopreserves thediagonal subgroup Handitsfrases'N(H),andthegroupsUandU~.TheWeylgroupofSp,,€can identified with thepermutations inS,,such that a(n+1)=of()£mfor ‘all<i<n,and itisexactly fortheseoforwhichonecanchooseamonomial PerntSP4C- Now ifisanyelement inSp,4C, writeg=w-n,"b ingtotheabove corollary. Then 9=90) =GU) (4): (6), 398 23.ComplexLieGroups:Characters and byuniquenessofthedecomposition wemusthave@(u)=1,4)=Mh, heHand9(b)=h"'-bUtollowsthatbelongstotheWeylgroupofSp3xCThisgivestheBruhatdecomposition, and,moreover,auniquedecomposition ofg€Spzx€intou-n,-b,withwinU(o)7Sp2x€Sincethislatterisanaffinespace, this shows that thecorresponding Bruhat cell inthesymplectic ag ‘manifold isanaffine space. Exactly thesame idea works fortheorthogonal groups SOC, byrealizingthemasfixedpointsofautomorphisms ofSL,,Cofthe form At» M™!A-!-M, with Mthematrix giving thequadratic form Note finally thatifWis theelement intheWeyl group thattakes each root toitsnegative, then B-W"-Bisadense open subset ofG,afactwhichis evident fortheclassical groups bytheabove discussion. The corresponding Bruhat cellXy. istheimage ofU~inG/B, which isalso adense open set.It follows thatafunctionofsection ofalinebundle onG/B isdetermined byitsvaluesonU~.Fortreatisesdeveloping representation theoryviafunctionson U>,see[N-S]or[Zel}The following exercise uses these ideas tosketch aproof ofClaim 2357thatthesectionsofthebundleL_,onG/Bformtheirreducible representationwith highest weight 2 Exercise 23,62*. (a)Show that sections sofL.., arealloftheform s(gB)= (6,f(a)), where fis«holomorphic function onGsatisfying ' S92) =AGF(Q)_ forallxeB. (b)Letn’€N(H) bearepresentative oftheelement W”intheWeyl group which takes each element toitsnegative Show that fisdetermined byils value at1. {(c)Show that any highest weight forfmust be2,and conclude that 11°(G/B, Ls theirreducible representation F,with highest weight 2 The holomorphic functions fofthis exercise arefunctions onthespaceG/U.Inotherwords,allirreducible representations ofGcanbefoundinspaces‘offunctionsonG/U.Thisisonecommonapproachtothestudyofrepresent tions,especiallybytheSovietschool,ef.[N-S},[Zel].Functions onG/Uform acommutative ring, which indicates howtomake, thesumofall theirreducible representations intoacommutative ring,Infact, fortheclassical groups, these rings arethealgebras S',S°, and5!con- structed inLectures 15,17,and 19,cf.[L-T]. They arealso coordinate tings{fornaturalembeddings offlagmanifolds inproductsofprojective spaces. LECTURE 24 Weyl! Character Formula ‘This lecture ispetty straightforward: wesimply sat theWeyl character formula in{241thenshowhowitmaybeworkedoutinspecieexamplesin$242.Inparticular, ‘rederiveinthecaseoftheclassicalalgebrasformulasfrthecharacterofgivenirreducible representation asapolynomialinthecharactersofcertainbasicones(either theaerating orthesymmetric powers ofthestandard cepreentaion forsl,C and thee analogoes forsp3,C and#05) The profs ofthe formula aredeered tothe folowing two lectures’ The techniques involved here areelementary, though thedeeerminantal formulasaefstlycomples,involvingallthealgebraofAppendixA AI: The Wel character formula {PA2: Applications toclassical Liealgebras andgroups §24.1. The Weyl Character Formula ‘Wehave already seen theWeyl character formula inthecase ofs1,€, andit isonereason why wewere abletocalculate somany more representations in thatcase. WesawinLectures 6and15that fortherepresentation P,=S,C*“ofSL,Cwithhighestweight4=51,L,,thetraceoftheactionofadiagonal matrix A¢SLC with entries x,...., x,isthesymmetric function called theSchurpolynomial 5,(x,,-..,X,).Thisincludedaformulaforthemultiplicities, which are the coefficients ofthe monomials inthese variables InordertoextendthisformulatotheotherLiealgebras, letustrytorewrite thisSchur polynomial inaway thatmay generalize. TheSchur polynomial is defined tobeaquotient oftwoalternating polynomials: Oy ungSieod=Ty 4 24,Weyl Character Formula ‘These determinants canbeexpanded asusual asasum over thesymmetric group &,,which istheWeyl group %®.Writing x,=e(L,) inZ[A] asinthe preceding lecture, andwriting(—1)"forsgn(W)=det(W)forWintheWeyl group, thenumerator may beexpanded intheform eg DET sohn=(= DEE, +—DL) =F (-New +od where wewrite JforEijL, and wesetp=E(n ~i)Ly. Our formula therefore takes the form LeHee +0) Chari =LY" 0) oo X(—1)"e(W(p) ‘Thedenominator isthediscriminant ACK1y22Xa)Tey=Leto ~e(L)). Thiscanbewritten interms ofthepositive roots L,—Lyi<j,as AlyoyXe)=Tedar Ly) e-Kly ~Ly) Note also that p=Elm—Ly=Ly+(Ly+Ly)to+(LyHot by) 1 =Pg -L) Which isthe sumofthefundamental weights, andhalf chesumofthepositive roots ‘Thesearetheformulasthatgeneralize totheothersemisimple Liealgebras. Foranyweight 4define A,¢Z[A] by Au=SNOW). ean Notethat4,isnotinvariantbytheWeylgroup,butisalternating: W(A,)= (-1)"A, forWe. The ratio oftwo alternating polynomials willbe invariant. ‘Theorem 24.2(Weyl Character Formula). Letpbehalfthesumofthepositive roots. Then pisaweight, andA,#i.Thecharacter oftheirreducible repre- sentation T,,with highest weight dis Chart) =“2, (wer) AI, The Weyl Character Form a ‘Theassertions aboutparepartofthe following lemma and exercise, which willalso beuseful intheapplications: Lemma 24.3. Thedenominator A,ofWeyl’s formula is 4,=[](lar%)~e(—a12) =eo)T].-e-a) =e-0) Ta) PROOF.Sincee(p)=e(5:4/2)=[](a/2}theequalityofthethreedisplayedexpressions isevident; denote these expressions temporarily byA.The key point istoseethat Aisalternating. For this, itsuffices toseethat Achanges signwhenareflection inahyperplane perpendiculartooneofthesimpleroots isapplied toit,since these reflections generate theWeyl group. This follows immediately from thefirstexpression forAand(a)inExercise 24.4 below. Now, bythesecond displayed expression, thehighest weight term that appearsinAise(p),whichisthesameasthatappearinginA,.Calculating 1/Aformally asin(24.5) below, weseethatA,/A isaformal sum¥m,e()thatisinvariant bytheWeylgroup,and,usingpart(c)ofthefollowingexercise, ithas weight0.AsinTheorem23.24itfollowsthatA,/Aisconstant;and,since ‘and A,have thesame leading term e(p), wemust have A,=A 0 Exercise 24.4%.(a)IfW=W,,isthereflection inthehyperplane perpendicularloasimple roota,showthatW(a,)=—a,,andWpermutes theotherpositiveroots. (b)With Was in(a),show that W(p) =p—a).Deduce that pistheele- ment in*such that p(H,)=2(p,a)A@%.2%)=|forexchsimpleroota, Equivalently, pisthesum ofthefundamental weights. Inparticular, pisa ‘weight. (0)ForanyW¥JintheWeylgroup,showthatp—W(p)isasumofdistinct positive roots. Deduce thatW(p) isnotintheclosure ofthepositive Weyl chamber. Proofs ofthecharacter formula willbegiven in§25.2 andagain in§262. Fornow weshould atleast verify that itisplausible, ie,that A,,,/A, isin Z{A]" andthat thehighest weight that occurs is4.Note that since the tumerator anddenominator atealternating, theratio isinvariant, The factthatA,isnotzerofollowsfromthesecondexpression inthepreceding lemma.Toseethat theratio isactually inZ[A], however, wemust verify that ithas only afinite number ofnonzero coefficients. Write aor 24,WeylCharacterFormula 1 Ps tee =e(- ay! =e, <n) (4HO)Tp eareteIT3,ene45) Whenthisismultipliedbyy,,=(—"e(W¥(2 +p)),wegetaformalsumwhere thehighest weight thatoccurs istheweight 2.This means inparticular thatthereareonlyafinitenumber ofnonzero termscorresponding toweightsinthefundamental (positive) Weyl chamber WButsince theratio isinvariant bytheWeyl group, thesame istrueforallWeyl chambers, 80A,,9/A, isinZLAJ™,andhashighestweight2.Itfollowsinparticular thattheAys/Ay,a8 Avaries over W’7A,formanadditivebasisforZ[A]™. Before considering theprooforanyotherspecialcase,weapply(WCF)to give aformula forthedimension ofTy: Corollary 24.6.Thedimension oftheirreducible representation Tis Btpe pyBtpa dim 0,=[]SP oopySTP A eres where (a,B)=a(H,) =2(c,B\MB, B)and(,)istheKillingform. Proor.Thedimension ofIP,isobtained byaddingthecoefficients ofalle(a)inChar({),ie,, computing theimageofChar(I,) bythehomomorphism fromZ{A] toCwhich sends each e(a)to1.However, asinthecase oftheSchur polynomial, thedenominator vanishes ifwetrytodothis directly. Toget around this, wefactor thishomomorphism through thering ofpower series: Z[A]* [tel] +€, where thesecond homomorphism setsthevariable ¢equal tozero, ic.picksofftheconstant termofthepowerseries,andthefirsthomomorphism takes e(a)toe**. More generally, foranyweight jtdefine ahomomorphism YeZEAJ—-CLLD},—ela)yverer, Weclaim that¥,(4,) ='P,(4,) forallLandj.This isasimple consequence ofthe invariance’of the metric, )under theWeyl group: (Ay)=I(—ream =S(-nrertonwe aE (-rere =¥4,) Therefore, (Ay) =¥,(4,) =Yay) =[en ay {242. Applications toCassia LiAlgebras and Groups wo: -(Le»)em4termsofhigherdegreeint Hence, V(Arsalan)=PAsAQ) =Uene)+termsofpositivedegreein¢, which finishes theproof a Exercise 247, Inthecase ofsl, verity that theabove corollary gives the dimension wefound inLecture 6. Exercise 248, Verily directly that theright-hand side oftheformula forthe dimension ispositive. Sinoe 11=Ausy/A,i8thecharacterofavirtualrepresentation which takes onapositive value attheidentity, a8inthecase offinite groups, to prove that iisthecharacter ofanirreducible representation, isulfices to show thatfo1s2s=IforanappropriatecompactgroupG.Thiswasthe ariginal approach ofWeyl, which wewilldescribe inthelastlecture. Since thehighest weight appearing is2,wewill know then that thisirreducible representation must beT3. Exercise 249. Use Corollary 24,6 toshow that if2is adominant weight Gx,intheclosure ofthepostive Weyl chamber) andois afundamental weight, then thedimension ofTi, isgreater than thedimension of Conclude that thenontrivial representations ofsmallest dimension must be mong themrepresentations f,with wafundamental weight. $24.2. Applications toClassical LieAlgebras and Groups Inthe caseofthe general linear group GLC, thecharacter ofthe represen- tation the Schur polynomial byte tfal "Wewetherepresentation ofGL,CiendfitrsctiontoSLC.sinetheaterwouldteasetheproductohevale0 os 24,WeylCharacterForma wich hasseveral expressions interms ofsimpler symmetric funtion Note that thecharacter ofthedihsymmetric power ofthestandard representa tion isthedthcomplete symmetric polynomial 11,innvariables (Appendix Ad} Hy=Char(Sym‘(C")). ‘The first “Giambelli” ordeterminantal formula (A.5) ofAppendix Agivesthecharacteroftherepresentation withhighestweight2=(1,>~~">2,>0)asanr xrdeterminant: Hy. Char Cy)=ages |ee |. (24.10) Aaa Ay Equivalently, thisexpresses ageneral element F,€R(G)oftherepresentation ringasapolynomial intherepresentations Sym“(C"*). Asecond determinantal formula, from(A.6),expresses I,intermsofthebasicrepresentations A‘(C*),whose characters aretheelementary symmetric polynomials Eq=Char(M(C*)). ‘This formula is,with gtheconjugate partition toA, Eg, Eyer Egytint Char(l) =LEgsed= [etn (ay Inthis section wework out thecharacter formula fortheother classical Liealgebras, including analogues ofthesedeterminantal formulas, Theana- logues ofthefirstdeterminantal formula (24.10) were given byWeyl, butthe analoguesof(24.11)werefoundonlyrecently({Ko-Te]).Wealsopay,atleast bywayofexercises, thedebtsto(WCF)thatweowefromearlierlectures. ‘The Symplectic Case ‘Theweightsforsp,,Careintegrallinearcombinations ofL,,...,Ly.Weoften writef=(sysensHa)FortheweightjyLy4°°"+pgLy ‘Thepositive roots are{L,—Ly},<jand{L,+Ly}.<).fromwhichwefind PHL OtlL Ly+(Ly+Lg)to +(LytoFL (24.12) ep =O Noo $242. Applications (oClascal LieAlgebras andGroups ws 4, | ‘Aswesaw inLecture 16,anelement intheWeyl group can bewriten uniquely asaproduct 60,where oisapermutation of(Ly... La} and =(6,-0.5GhWith6=+1.Hence: ALVES re “an e419) here thesign (~1)' istheproduct ofthec,.Now with x,=e(L,), thiscanbe written © A=LY] tinxa) or Ayala 44 where|a,,|denotesthedeterminant ofthenxmatrix(a,,)Inparticular, Ay=bop xp tony, (24.15) From (2414) orExercise A.52 wehave Ag=MK,XToesXetTOETDS ee" (24.16) where Aisthediscriminant. Exercise 24.17, show that Aa=[]ersereg~DTDGe=Derra) ‘The character oftheirreducible representation F,with highest weight dnSebibedy2-22,2Ovithereto: 406 24,WeylCharacterFormula papitata ty Chatty) =rar caret (24.18)a al ThedimensionofI)iseasilyworkedoutfromCorollary24.6: =. Wh dime= OLY.QG=0 aree-i=5 WY) yh pats Tot=mTi my WhereI,=A, +n— 1+Land m=n—i+L Exercise 24.20, Show that, setting Ij=4+~i, LG D+G+TG+9 im(t) =SP 8)=Gan=Ht ‘Theseformulasgivethedimension oftheirreducible representation T,,,..«with highest weight 4,00, +"+ ayy where the«,arethefundamental weights, using therelation ay=a)+"°""+ ay. Exercise 24.21. Use Exercise 2420 toverify that for A=Ly+°**-+ Ly,the dimensionofTyis2nifk-=I,and()-(*)ifk=2,Usethisto giveanother proof thatthekernel ofthecontraction from MVtoA-4Vis irreducible. ‘The firstdeterminantal formula forthesymplectic group goes asfollows Let We 005%) Hey eyXueTones Bat Where His thedthcomplete symmetric polynomial in2nvariables. Inother words, J,ithecharacter oftherepresentation Sym“(C™) ofsp2,C. From Proposition A.50 ofAppendix Awehave Proposition 2422. If4=(A2 >A,>0)thecharacter ofTisthe deter inant oftherxrmatrix whose ithrowis Cain Suctea tit Sates tact oeSagan tSiptesah Forexample, for4=(d)ie,A=dLy, wehave Chat(Cy)) =Jy,which isthecharacterofSymé(C"), Inparticular, thisverifiesthatthekthsymmetricpowers Sym*(C?) ofthestandard representation areallirreducible. (This, of course isaspecial ease ofthe general description given in§17.3, since allthe contraction maps vanish onthesymmetric powers) 242. Applications toClassical LieAlgebras andGroups “07 Exercise24.23,(i)Findthecharacteroftherepresentation ofsp4Cwithhighest weighto+02=2L,+La,verifyingthatthemultiplicities areaswefound in§16.2. (i)Find thecharacter oftherepresentation ofspqC with highest weight wy+a,thus verifying theassertion ofExercise 17.4 ‘Thesecond Giambelli formula inthesymplectic caseexpresses Fin terms ofthebasic representations Ta,=Ker( (0) +A-4(C%)) which arethekernels ofthecontractions, Thecharacter ofI,isEj,whereByeLEBextetaybay!botxyand BC eyaeXTyeyMat)—EnahyyeMTooa) fork 22,where Eyisthe kthelementary symmetric polynomial. Theformula is Corollary 2424, Letj=(sy,..-44)betheconjugate partition to2.The‘characterofTisequaltothedeterminant oftheIxImatrixwhoseithrowis (Ein Enmtea +Beet Eyouns +Bait Egat +Catia) Proor. This follows from theproposition and Proposition A.44, which equates thetwodeterminants before specializing thevariables. fal ‘Thereisalsoasimpleformulaforthecharacterintermsofthecharacters £,ofAY(C™),whichalsofollowsfromProposition A.44: Char(F) =1Bts) ~Burt-ah (24.25) Note that Eysy =Ez» (corresponding totheisomorphism At**C?* = N-*C™) and” Exe —Ey-ava- Inparticular, Corollary 24.24 expresses Char(F;) asapolynomial inthecharacters ofthebasic representations T,, woePay ‘TheOdd Orthogonal Case Forsozeyi€ theweights are3Lyj=(js++a)Withallintegersorall, halfintegers.Thepositiverootsare(L—Ly}:ej{Lr+L,)s<j,and (L,},$0p is(Ly +++ +L,)lessthan inthecaseforsp3. p=Linth— DL, (24.26) or pa(n— dn Food With xf!=(£1) andxf! =e(4:1,/2), wehave thesame formula as before [(24.14)] forAy 408 24,WeylCharacterFormula Exercise 24.27*. Show that Apabap =ety AG,HGoyheHME)Oe?=THY.Gel?=xp) ITyistheirreducible representation with highest weight 4=SAL), A,2+" 24,20,then thecharacter formula canbewritten Chany =poy aunac(T)=roe Similarly, i WD +)dimer) = [=P Gtime)=Gyowst=i=p =A Ah =phin’, 2429 Hen=miy Mn eid wherel=A,-+n—i+ $,andm=n—i+$. Exercise 24.30. Show that, with jj=4,+ —i, TG DG45+OTTAK+O im(t) =F oi=Gn Gn= ‘These formulas givethedimension oftheirreducible representation T,,...«, with highest weight a, +*-*-+ a,0,, where the«,arethefundamental weights, using theequations oe es Exercise24.31.Usethedimension formula toverifythatforA=Ly++"+Lys thedimension ofFis(*ft‘)Usethistogiveanotherproofthat/*Vis irreducible for|<k<n.Verifythatthedimension ofthespinrepresentation is2",thus reproving that itisirreducible. Exercise 24.32. Use thedimension formula toverify that thekernel ofthecontraction :Sym‘(C™") -+Sym?2(C2%1) isanirreducible representation with highest weight dL. Incase therepresentation isarepresentation ofSO,,4,©;i¢,theA,areall- integral, there isafirstdeterminantal formula that expresses I’,interms of the kernels ofthe contractions Ker(Symmt(C™"")-» Symé-4 C2"), 242. Applications toClassical LieAlgebras andGroups 409 LetK,denotethecharacterofthisKernel,s0Ke=I,Ky=x,bo$4ailfetag! +fand KgHA05oesSeeXiyeeBeWSHealy coeXaXioonesOe where H,isthedthcomplete symmetric polynomial. From Proposition A.60 wehave Proposition 2433. IfA= (Ay2°": >2,>0), with theA,integral, then thecharacterofTisthedeterminant ofthe'rxrmatrixwhoseithrowis (Kater Kanter +Kat Kates +Kart Katee +Ka-iorea): Inparticular, for4=(d),thecharacterisK,,whichverifiesthatthekernel ofSym*(C2**1) -»Sym*-#(C™"1) isirreducible. Exercise 24.34. Usethecharacter formula toverify thatthemultiplicities of therepresentation Ty,,,ofSosC areasspecified inExercise 189. Thesecond determinantal formula forSOsq4,€ writes 1interms ofthe representations AM(C™*), whose characters are By=ElaoesXapXTyeehasDh Applying Proposition 24.33 with Corollary A.46, wehave Corollary 2435. Letj=(jy, ...444) betheconjugate partition toA.The characterofTisequaltothedeterminant ofthexImatrixwhoseithrowis Egetsn Enter +Epet Eger +Bgyt-iead Since Eyiy =Exs1-x (Corresponding tothe isomorphism A'**C2*! x NACH, thisexpresses Char(T) asapolynomial inEy,....Eywith E,= Char(M'c™*"), TheEven Orthogonal Case Forso4€theweightsarethesameasintheprecedingcase.Thistimethe{L} atenotpositiveroots,however,sopis(Ly+++"+L,)lessthaninthecase Of02441C,OFLy+°**-+LeylessthaninthecaseofsP34C: p=Ln- IL, (2436) or pa (n— Bn 2.40) Thecalculation ofA,issimilar,butusingonlythose«ofpositivesign.This time 410 24, Weyl Character Formula Lee festa)=4fon+8+ffon—8} This leads 10 A,=Mp tg" +bs". (2437, Note that thesecond determinant term vanishes when any j1,iszero. In particular, Ay Axl x, (2438) From (24.14) orExercise A.66, Ay=AUK,AToehtED (24.39) ‘This gives, with Ttheirreducible representation with highest weight AmDalyAy2°lal20, btty+bef— Char() =hey (24. Aaaa oe where I,=2,+n —1.Asbefore, , da) ey)dimr) = JP.its)=TN=)Gath aa“na, aaTht=mpi ean where I,=Ay+—dandm, =n—1.Note that,asexpected,thetworepresen tations with weights (2,,..., 4,1, +4,) have thesame dimensions. Exercise 24.42. Show that ee Se)a Hn =HB ‘These formulas givethedimension oftheirreducible representation T,...4. with highest weight a, +--+ 4,c,, where thea,arethefundamental weights, using theequations Amatotea+MG,4+a,TStisn—2, Baga =Myo $y) y= Haya +5). Exercise 24.43,Usethedimension formula toverifythatfora=Ly+°°>+Ly k<n,thedimension ofT,is(7).soAMC")isirreducible. For AeLy+++Ley£bythedimensionis(2).50AY(C)isthesumof thetwocorresponding irreducible representations. Verify that thedimension ofthetwospin representations are2"~', proving irreducibility again, {242. Applications toClassical LieAlgebras andGroups au ‘Note that thesecond term inthenumerator in(24.40) changes sign when Axis replaced by—4,; inparticular, itvanishes when 2,=0.When A,=0, therepresentation 17isarepresentation oftheorthogonal group Oz4C ‘When 2,#0,thedirect sum ofthetworepresentations with highest weights(yy--o5£4,)isanirreducible representation ofOgyC.(SeeExercises23.19and23.37) Let Lybethe character ofKer(Sym‘(C%)—> Sym*-7(C™), ic, LegHabeetyoopSupXiyooXu")—HaalpseosXapXTyooXe")Ieither «ase, Proposition A.64 applies togive thefirstdeterminantal formula: Proposition 24.44,GivenintegersA,>--->2,>0,thecharacteroftheirreduc- iblerepresentation ofOngwithhighestweightA=(Ay,...y4,) isthedeter-Iminantoftherxrmatrixwhaseithrowis (LayerLaetes+bayeroneRayoter+Layctorea) Again, for2= (4),this verifies that thekernel ofthecontraction from Sym*(C™) toSym*-*(C%) isirreducible.‘Theseconddeterminantal formulaisthesameasintheoddcase,butwithEyFultsooosSanMtoeMSF Corollary 2445. Lety=(sty... 44)betheconjugate partition toA.The characterofTisequaltothedeterminant ofthexImatrixwhoseIthrowis CEgetss Eyy-teat Byte Egy-trt Eyetteal Using thefactthat Eyy4 =E,-4. thisexpresses Char(T3) as@polynomial inEyyoooyBywithEy=CharMC). Exercise 24.46*. For each oftheorthogonal groups O,€, show that the characteroftheirreducible representation withhighestweight4canbewritten intheform Char(Py)=tastes—Bayatsh Where isthecharacter ofSym*(C"). Another formula forthedimension of T,isobtainedbysubstituting (7)forhyinthisdeterminant. ‘There areother formulas expressing thecharacters ofgeneral representa-tionsintermsofsimplerones.Abramsky,Jahn,andKing[A-J-K]giveone that canbeexpressed bythesame formula forthegeneral linear, symplectic, andorthogonal groups. Thegeneral irreducible representations aregiven by partitions 2orYoung diagrams, andintheir formula thesimpler represen- {ations arethose corresponding tohooks. Toexpress it,let(ab) denote thehook with horizontal legoflength a+1 and vertical legoflength b+1,ie,thepartition (a+1,1,...,1), with &I's.More generally, given=(a,>">a,20)andb=(by>">b,20)witho,orb,nonzero,let (a+b) denote thepartition whose Young diagram haslegsofthese lengths (0 an 24,Weyl Character Formola therightofandbelowtherdiagonal boxes(cf.Frobenius’s notation, Exercise4.17).Lettug.s,denotethecharacter ofthecorresponding irreducible representation. Their formula is Leah =eight stser (any Taking thedegree ofboth sides gives newformulas forthedimensions ofthe irreducible representations. These formulas areparticularly useful iftherankrrofthepartitionissmall. Exceptional Cases ‘Wewill, asalastexample, work outtheWeyl character formula fortheexceptional Liealgebrag,,andtherebyverifysomeoftheanalysisofitsrepresentations given inLecture 22.The remaining four exceptional Lie algebras wewill leave asexercises.‘Tobeginwith,thevalueofpiseasilyseentobe2L,+3Lz,intermsofthe basis L,,Lyfortheweight lattice introduced inLecture 22. aaciea Now, foranyweight =pL,+qL2+rLy, wehave Ay3,fayhayXen YeXelaXda =BOIS apl8)=BONS,«0°, $242.Applications toClassicalLieAlgebrasandGroups as wherewewritexor(x,X353)andx7"for(x7,x51,x5")Aisthediserim- inant,andS,,.,theSchurfunction.Usingtherelation[]x,=1wecanalso write this as =BE) (Sy.g06) ~Sxepim-ain el) forany m2 max(p, 4,1)Tomake thisnotation agree with thestandard notation forSchur polynomials from Appendix (A.4), note that5,4, isthe Schur polynomial S,.»forthepartition (s,s >&,where sistwofessthanthedifferencebetweenthelargestandsmallestofp,q,andr,whileisonelessthan thediference between thesecond largest and thesmallest; ifp,q, and r arenotdistinc, 5,,=0.Thus, forexample, Ay=BG)(S1,4)=Sy) =ACs) (xe +4%5 40%) —44—2—43) Now, anyirreducible representation Tofg,hashighest weight 2=aor, + borg, where 0,=Ly+Lyand a;=Ly+2L; are the two fundamentalweights,andaandbarenon-negativeintegers. Then4+p=(a-+b+2)Ly+ (a-+2b+3)L,.TheWeylcharacterformulainthiscasebecomes Proposition 24.48.Thecharacteroftherepresentation of3wthhighestweightfay+boois Char(Ty,,) =Seszeeteseen =Sies2bsinSau Say Exercise 2449. Inthecase ofthe standard representation To theadjoint representation Tp,,andtherepresentation T, usethisformula toverily the multiplicities found inLecture 22, Wecanalso work outthedimension formuta explicitly inthis case. The twofundamental weights and «,have inner products (o,e)=1 (yo)=32and—(04)=; ‘0,and a,areamong thepositive roots ofgz,and interms ofthese thetemainingpositiverootsate20,~ay,30;~0g,0;~04,aNd200,~3,Theweightpisthesumofthe fundamental weights«»,ando,,sothatforan arbitraryweight4=ac,+borwehavethefollowingtableofinnerproducts: A bd Cato Too, 12a (+n dy—0, 3 NaF IA Hab+2y2 ‘a,5h a44hp ah4592 = 92 ged Mas 4-H <0; 2 4% (as 3442 to, +20, 32 he x42 a4 24,Weyl Character Formala ‘Weconclude thatthedimension ofthe irreducible representation Tyofgywith highest weight 2=aor +boos is int,,)=(@tDA+B+2020+3b+Hfa-42+Ia+364441) a a Wecancheck this inthecases a= 1,b=0and a=0, b=1,getting the dimensions7and14ofthestandardandadjointrepresentations, respectively. Incase a= 2, b=0wemay verify theresult oftheexplicit calculation in Lecture 22,finding that im(T 9)=27 and,therefore, deducing thatA?V =T3,o@ V@€andSym*V =T;,o@C. Exercise 2450. Show thatSym*V =@HYT-24,0- Weleavetheanalogouscomputations fortheremainingfourLiealgebras asexercises,usingthedescription oftherootsystemsfoundinExercise21.16.Since wehave notsaid much about theWeyl group intheexceptional casestheformula(WCF)cannotbeuseddirectly—nottomentionthefactthatthe‘orders ofthese Weyl groups are:27-3? =1152 forfa;27+3*+5 =SI,840 for egy219-3457 =2,903,040 for€5,and2!*:3%+5?-7 =696,729,600 forty. However, thedimension formula isavailable. Exercise 2451*. For each ofthefour remaining exceptional Liealgebras,computep=halfthesumofthepositiveroots.Foreachofthe fundamental ‘weights«,atleastforf,,computethedimension oftheirreducible represen-tation with highest weight «.Inparticular, find thenontrivial representation ‘ofminimal dimension. Usethistoverily that (Eg) isnotisomorphic to(By)(oF(Cy),itathat¢gisnotisomorphic to$0;5€osp,2€. Exercise24.52°.Listallirreducible representations VofsimpleLiealgebras‘such that dim V<dim g,Note that these include allcases where thecorre- sponding group representation hasaZariski dense orbit, orafinite number oforbits. LECTURE 25 More Character Formulas Inthis lecture wegive vo more formulas forthe muliplces ofan irreducible representation ofa semisimple Liealgcbra orgroup. First, Freudenthal's formula (25.1) gives astraightforward way ofcalculating themultiplicity ofagivenweight ‘onceweknowthemultiplicity ofallhigherones.Thisin(urnallowsustoprovein 252 theWeyl character formula, aswell asanother multiplicity formula due to Kosta. Finally in§25.3 wegive Stenberg’ formula forthedecomposition ofthe {tensorproductoftwoarbitrary irreducible representations ofasemisimple Liealgebra, andalso giveformulas forsome pairscgforthedecompesiton oftherestition to'ofreducible representations of £254: Freudenthats moliplicty formula {252: Proof ofWSF}, theKestant multiplicity forma£253:Tensorproduceandrestrictionstosubgroups §25.1. Freudenthal’s Multiplicity Formula Freudenthal’s formula gives ageneral way ofcomputing themultiplicities ofarepresentation, ic,thedimensions ofitsweightspaces, byworkingdown successively from thehighest weight. The result issimilar to(but more complicated than) what wedidforsl;C inLecture 13,where wefound the multiplicities along successive concentric hexagons intheweight diagram. LetTy,betheirreducible representation with highest weight 2,which will befixedthroughout thisdiscussion. Let, =nF) bethedimension ofthe weight space! ofweight yxinTy,i¢.,Char(T,) =J-n,e(1). Freudenthal gives aformula forn,interms ofmultiplicities ofweights that arehigher than j. "antheerste map arefen referred Yo8ner mice” 416 25.MoteCharacterFormas Proposition 25.1 (Freudenthal's Multiplicity Formula). With theaboe notation, ea)mf)=2FE (Htkag oan where e(1) =1+ pl?~la+el?. HereIPI?=(B,B),(,)istheKillingform,andpishalfthesumofthepositive roots. Exercise 25.2°. Verify that c(u) ispositive ify#Aand n,>0. ‘The proof ofFreudenthal’s formula uses aCasimir operator, denoted C. ‘This isanendomorphism ofanyrepresentation Vofthesemisimple Lie algebra g,andisconstructed asfollows. Take anybasis U,, ...,U,forg,and letUj, ...,Usbethedual basis with respect totheKilling form ong.Set CHU,U + +UU, ic,foranyveV,CW)=YU, (Uj-v). Exercise 25.3,VerifythatCisindependent ofthechoiceofbasis”, “The keyfact is Exercise 25.4*. Show that Ccommutes with every operation ing,ie, C(X-v) =X-Cwv) forall Xeg.ve Vv. ‘Theidea istouseaspecial basis fortheconstruction ofC,sothateach term U,U; willactasmultiplication byaconstant onanyweight space, andthisconstant canbecalculated intermsofmultiplicities. ThenSchur’slemmacanbeappliedtoknowthat,incaseVisirreducible, Citselfismultiplicationbya scalar. Taking traces willlead toarelation among multiplicities, anda little algebraic manipulation willgive Freudenthal's formula. ‘Thebasisforgtouseisanaturalone:Choose thebasisH,,..., H,fortheCartansubalgebra, whereH,=H,,cortesponds tothesimplerootay,andkt H;bethedualbasisfortherestriction oftheKillingformtoh.Foreachroota,chooseanonzero X,€g,.ThedualbasiswillthenhaveX}ing_,.Infact, ifweletY,€q-,betheusual element sothat X,,¥,,andH,=[X,, ¥,)are thecanonical basis forthesubalgebra s,=s1€ thal they span, then XL= (02% as) Exercise 25.6*. Verify (25.5) byshowing that(X,, ¥,)=2/(a, a). “tnlancy language,eaelementoftheuniversenvelopingalgebrafgbuwedonotnet ‘me {25 Freudenthat's Multiplicity Formula an Now wehave theCasimir operator C=LMM+YX andweanalyzetheactionofContheweightspaceV,corresponding toweight for anyrepresentation ¥.Letn,=dim(V,). First wehave Ai; actson¥,bymultiplication by(1,2)=lyl?. (25.7) Indeed, H,Hj actsbymultiplication byj(H,)y(H)). Ifwewrite p= Yona, where theaarethefundamental weights, then j(H,) =7,andif= Sri, with«thedualbasistoa,thensimilarlyu(H;)=rj.HenceY)n(H)u(H;) =Sel =(i1),asasserted. Now consider theaction ofX,Xz =((2)/2)X¥,onF,,Restrictingtothe subalgebras,=sl,andtothesubrepresentation @))¥,.1,corresponding to thea-stringthrough1,weareinasituationwhichweknowverywell.Suppose thisstringisVy@VpeB" Vpn ‘80m=fi(H,) (ef.(14.10)]}, and letkbetheinteger such that j= f—ka.We assume fornow that k<m/2. ‘Onthefirst term Vy,X,¥, acts bymultiplication bym=f(H,) =2B,a)(x,a),soX,X;actsbymultiplication by(f,2).Ingeneral,onthepartof¥s-,, which istheimage ofVgbymultiplication by(¥,}, weknow [el(11.5)]} that X,¥,acts bymultiplication by(k+1)(m —k).This gives usasubspace of¥,ofdimension n,onwhichX,X7actsbymultiplication by (+ 19(B, 2)—Ka29/2) mk +YC 2)+RC,2/2. ‘Nowpeeloffthesubrepresentation (overs,)ofVspanned byV,andapply thesame reasoning towhat isleft.Wehave asubspace ofV,_, ofdimension14-4—ty(0whichthesameanalysiscanbemade.Fromthiswegetasubspace of¥,ofdimension mp_,—nyonwhichXX;actsbymultiplication by (U4 2)+(k~(a, 292). Continuing topeeloffsubrepresentations, thespace V,isdecomposed into pieces onwhich X,X; acts bymultiplication byascalar. The trace ofX,X; onF,istherefore thesum + DEL a)+Hea2) +G0 9)COUUH +Mee 2 FH pete ~Maer) MC 2)+(0) 99/2). Canceling insuccessive terms, thissimplifies to ‘TraceXaXalr,)=J(H+itspee: (258) ‘Onepleasant factaboutthissumisthatitmaybeextended toalli>0,sinceMya=Ofori>k, a8 25, Mote Character Formulas Incasek>m/2,thecomputation issimilar, peelingoffrepresentations from theotherend,startingwithV;-..Theonlydilferenceisthattheactionof u%, 0MVp i260, Theres 8 TraceKK)=7S0,ay 259) Exercise 25.10, Show that X,X; =A.A, +(a,4/2)ll,, anddeduce (255) directly from (25.8) byreplacing 2by—a Infact, (25.8) isvalid forall4and a,aswescefrom theidentity Ee+tmalga=0. es) Exercise25.12*.Verify(25.11)byusingthesymmetry ofthe a-string through p. Now weaddtheassumption that Visirreducible, soCismultiplication bysome scalar c.Taking thetrace ofCon V,andadding, weget on,=(Hm +FY(at+iaOVote (25.13) Note that when i=0thetwo terms for«and —acancel each other, sothe summation canbeginati=|instead,Rewriting thisintermsofthepositiveweights, andusing(25.11)thesumsbecome XY +iaeet FYUeiepate myF042 YSUttates ‘Summarizing, andobserving thatJeas(ia)=(1,2p),wehave er,=(ie#)+(ie20)+2YeYH+IMpa Note that (1,1) +(94,20) =(+ p,+p)~(p.) =hae+pl?—Bp. ToevaluatetheconstantweevaluateonthehighestweightspaceV;,wheren,=1 and,,., =0fori >0.Hence, =(4)+(2p)=1+oP—WP. sia) ‘Combining thepreceding twoequations yields Freudenthal’s formula. Exercise 25.15. Apply Freudenthal’s formula totherepresentations ofsl;Cconsidered in§13.2,verifyingagainthatthemultiplicities areasprescribedonthehexagons and triangles. §25.2. Proofof(WCF);theKostant Multiplicity Formula alo Exercise 25.16. Use Freudenthal’s formula tocalculate multiplicities forthe representations I,0,T9,1,andTy,9of(93). §25.2. Proof of(WCF); theKostant Multiplicity Formula Itisnotunreasonable toanticipate that Weyl's character formula canbe deduced from Freudenthal’s inductive formula, butsome algebraic manip- ulation iscertainly required. Let a= Char(ly) =Fnye) bethecharacter oftheirreducible representation with highest weight 2. Freudenthal’s formula, inform (25.13), reads* eneLovamead +E&(1+in,2)see, where c=[2+pl?—fp. Togetthistolook anything likeWey!’s formula, ‘wemustgetridoftheinsidesumsoveri.If«isfixed,theywilldisappear if \wemultiplybye(2)—1,assuccessive termscance!: (ola)~FF+aamped =Fisage+a LetP=[een(e(a)—1)=(e(a)~1)-P,,whereP,=[]pra(e(B) ~1).Thereceding twoformulas give Pet =PS (ussdmyea) +Y(HuadPangetse +a). (25.17) Note also that Pm(-1Ay Aye where risthenumber ofpositive roots, soatleast theformula now involvestheingredients thatgointo(WCF). ‘Wewant (oprove (WCF): A,11=Ais».Wehaveseenin§24.1thatboth sides ofthisequation arealternating, andthat both have highest weight term e+ p),with coefficient 1.Ontheright-hand sidetheonly terms that appear arethose oftheform +e(W(A +p)), forWintheWeyl group. Toprove (WCF), itsuffices toprove that theonly terms appearing with nonzero coefficients inA,-z, arethese same e(W(A +p)),forthen thealternating property and theknowledge ofthe coefficient ofe(4+p)determine allthe coefficients. Thiscanbeexpressed as: nhssectionweworkinthrngCCA]itesomsYnwitcompleoicientm- 40 25. Mote Character Formulas Claim, The only terms (x)occurring InAy-xs with nonzero coefficlem are those with Wh=A+ pl. ‘Toseethat thisisequivalent, note that bydefinition ofA,and y,,thetermsinAy,atealloftheform+e(s),wherev=1+W(p),foreaweightofF,and WintheWeyl group. ButifIj.+W(o}l =[4+plsince themetricis invariantbytheWeylgroup,thisgives|W~'(x)+pl=JA+pil.Butwesaw inExercise 25.2 that thiscannot happen unless x=W(A), asrequired. Wearethus reduced toproving theclaim. This suggests looking atthe “Laplacian” operator that maps e()tofyel?e()0, that is,themap A:C[A] +C[A] defined by ACEmend) =F(a wegen). ‘Theclaimisequivalent totheassertionthatF=A,”z,satisfiesthe“differential equation” A(F) =14+pF. Fromthedefinition A(x.)=(v4)nge(u).AndA(A,)=pl?4,.Ingeneral,since |W(a)l| =YallforallWe WB, ACA.) =E(DLW (@)e(W (2)=La?4,, ‘Sowewould beingood shape ifwehadaformula forAofaproductoftwo functions. One expects such aformula totake theform Ua) =Aa +208f, Va)+JACO) 2518) where Visa“gradient,” and (,)isan“inner product.” Takingf=e(u g=e(v),weseethatweneedtohave(Ve(),Ve(»))=(4,ve(u+¥).Thereis indeed such agradient and inner product. Define ahomomorphism V:CLA]-+§*@CLA]=Hom(h,CTAI) bytheformula V(e()) =s-e(j)), and define thebilinear form (,)on '°@CLA] bytheformula (acts), Be(o)) =(a,Ble(u +v),where (x, isthe Killing form on§*. Exercise 25.19. With these definitions, verify that(25.18) issatisfied, aswellas the Leibnitz rule Visa) =Vif) +S¥(0). Forexample, ¥(z1) =Eynyst- es)and,bytheLeibnitz rule, Ver)=5Pexelah But now look atformula (25.17). This reads $252. Proofof(WCFtheKostantMultiplicity Formala a Pra =PAU) +(VP, Vita) Since, also bytheexercise, V(P)=2(—1YA,V(A,), wemaycancel(—1YA, from each term intheequation, getting ©Agta =ApA(ts) +2(VAys Vis) Bytheidentity (25.18), theright-hand sideofthisequation is (4,23) —(A,)2=BA,24)~WPA,Re Sincee =|+pl?~pI?thisgivesI+pl7A,x,=A(A,x2)whichfinishes theproof. Bb Weconclude thissection with aproof ofanother general multiplicity formula, discovered byKostant. Itgives anelegant closed formula forthe multiplicities, butattheexpense ofsumming over theentire Weyl group (although aswewillindicate below, there aremany interesting cases where all butafewtermsofthesumvanish). Italsoinvolvesakindofpartitioncounting function. For each weight 1,letP(x) bethenumber ofways towrite asa ‘sumofpositive roots;setP(0)=1.Equivalently, 1 ty Pel 5.Jaw 7Ee (25:20) Proposition 25.21. (Kostant’s Multiplicity Formula). Themultiplicity n,(U)of weight jintheirreducible representation I,isgiven by nt)=1)"PW+p)—(e+ rE)=3(POE+9)—(u+ where pishalf thesum ofthepositive roots. Proor. Write(A,)"! =e(—p¥[] (1—e(—a))=5,P(vle(—v —p).By(WCF), a>AseelA,) =F(NEE +pPe(—¥ —0) =ECprpoe wea+9)-+0) =ECVtrna +a)—(+Met, asseen bywriting p=I(A +p)—(v +ph. o Infact,theproofshowsthatKostant’s formula isequivalent toWeyl's formula, ef.[Cart]. ‘One way tointerpret Kostant’s formula, atleast forweights 1close tothe highest weight 4ofI,isasasort ofconverse toProposition 14.13(ii). Recall that thissays that Iwillbegenerated bytheimages ofitshighest weight vector 0under successive applications ofthegenerators ofthenegative root spaces;inpractice, weusedthisfacttoboundfromabovethemultiplicities of an 28, More Character Formulas various weights close toAbycounting thenumber ofways ofgetting fromAtobyaddingnegativeroots.Theprobleminmakingthisprecisewasalways that wedidnotknow how many relations there were among these images, if any. Kostant’ formula gives ananswer: forexample, ifthedifference A—tissmallrelative(oA,weeethattheonlynonzeroterminthesumistheprincipleterm,corresponding toW=1;inthiseasetheansweristhattherearenorelations other than thetrivial ones X(¥(o)) ~Y(X(o)) =LX,¥1(o). Whengelssomewhat smaller,othertermsappearcorresponding tosinglereflectionsWin thewalls ofthe Weyl chamber forwhich W(A +p)ishigher than y+p;wecanthinkofthese terms, which allappear with sign —1,correction terms indicating thepresence ofrelations. Asygets smaller sill, ofcourse, more terms appear ofboth signs, andthisviewpoint breaks down.Toseehowthisworksinpractice,thereadercanforexamplecarryouttheanalysisoftheexampleattheendof§13.1 Exercise 25.22* (Kostant). Prove thefollowing formula forthefunction P, Which canbeused tocalculate itinductively: P(0) =1,and, for #0, Pn) — FCP +W)~ Exercise 25.23* (Racah), Deduce from Kostant’s formula andthepreceding exercise thefollowing inductive formula forthemultiplicities m,ofpinTrin,=Vif=A,andifeisanyotherweightofF,then ==L(KWhgep arn 9=F(HDTneyarr Show, infact, thatforanyweight jt og rpm =EHV wherethesecondsumisoverthoseW’eWBsuchthatW"(A+p)=1+p. NotethatKostant’sformula,morethananyoftheothers,showsusdirectly thepattern ofmultiplicities intheirreducible representations ofsl,C. For ‘one thing, itis easy torepresent thefunction Pdiagrammatically: inthe Weight lattice ofsl,C, thefunction P() will beaconstant 1ontherays {aL ~aLy},z0 and{als —aLz},2o through theorigin inthedirection of thetwosimple positive roots Ly—L, andLy—La.Itwillhave value 2.00 thetranslates {aL —(a+3)Ly}45-y and(aL, ~(@ ~3)L;}epofthese(wo raysbythethirdpositiverootLy—Ly:forexample,thefirstofthesecanbewritten as Ly~(a+3)Ly=(a+1)(Ly~Ly)+Ly~Ly =(a4 2)-(Ly—Ly)+LyLas and correspondingly itsvalue willincrease by1oneach successive translaleoftheseraysbyLy~Ly.Thepictureisthus §25.2. Proofof(WCF);theKostant Multiplicity Formula. 423 naePRR” Now, theprescription given intheKostant formula forthemultiplicities isto take sixcopies ofthisfunction Mipped about theorigin, translated sothat ‘thevertex oftheouter shell liesatthepoints w(A +p)—p and take their alternating sum. Superimposing thesixpictures wearrive at iNreehpesKOO)NSNW whichshowsusclearlythehexagonal patternofthemultiplicities. Exercise 25.24*. Anonzero dominant weight 4ofasimpleLiealgebra iscalled minuscule if(H,) =0or|foreach positive root a. fa)Show that ifAisminuscule, then every weight space ofTFisone Ronen au 25, More Character Formulas (b)Show that Jisminuscule ifand only ifalltheweights ofF,areconjugate under theWeyl group. {c)Show that aminuscule weight must beone ofthefundamental weights. Findtheminuscule weights foreachsimpleLiealgebra. §25.3. Tensor Products andRestrictions ToSubgroups Inthecase ofthegeneral orspecial linear groups, wesawgeneral formulas fordescribing howthetensorproductI,@[,,oftwoirreducible representa-tions decomposes: TOT, =Ql Inthese cases themultiplicities N,,, canbedescribed byacombinatorial formula: theLittlewood-Richardson rule. Ingeneral, such adecomposition isequivalent towriting at Lage @529 inZ[A], where x,=Char(P,) denotes thecharacter.* ByWeyl's character formula, these multiplicities N,,, aredetermined bytheidentity Arse Ane =ONip’ Arty 2528) Thisformulagivesaneflectiveprocedure forcalculating thecoefficients Nyy,ifonethatistediousinpractice: wecanpee!offhighestweights,ie,successively subtract from A,,p° 4,«multiples of4,A,,,forthehighestvthatappears. There aresome explicit formulas fortheother classical groups. R.C.King [Ki2] hasshowed that forboth thesymplectic ororthogonal groups, the multiplicities ,,,aregiven bytheformula Nig=3Mees’MenMover (2527) wheretheM’sdenotetheLittlewood-Richardson multiplicities, i¢.,thecorre:sponding numbers forthegeneral linear group, and thesum isover all partitions {,o,t.Forother formulas fortheclassical groups, see[Murl), {Wel,p.230]. Exercise 25.28*.Fors04C,showthatallthenonzeromultiplicities N,,,areI's,andtheseoccurforvinarectanglewithsidesmaking45°anglestotheaxes.Describe this rectangle “tnthetterature thesemultiples W,,areoftencalled“outermultpcies“and theeoblewolfindingtheodecompoting theenorprodct,the“ClebachGordon”problem, {§253. Tensor Products andRestrictions toSubgroups 425 Steinberg hasalsogiven ageneral formula forthemultiplicities N,,,. Since itinvolves adouble summation over theWeyl group, using itinaconcrete situation may beachallenge. Proposition 25.29 (Steinberg's Formula). Themultiplicity ofU,inT,® Fis Naps YD" POA p)4WW+A)—¥~20) wherethesumisoverpairsW,W"€48,andPisthecountingfunctionappearing‘inKostant’s multiplicity formula. Exercise 25.30°, Prove Steinberg's formula bymultiplying (25.25)byA,,using (WCF) t0getZeAysy =DNagyAvsp: Write outboth sides, using Kostant’sformulaforx4,andcomputethecoefficient oftheterme(f+p)oneachside,forany f.This gives Dy(DM POVEA+p)+WH+0)~B20) =DOWN nmasore: Show that forf= valltheterms ontheright arezer0 butNay. Exercise25.31(Racah).UsetheSteinbergandKostantformulastoshowthat Nagr=F(DMs grrnlTa Thefollowingis thegeneralization ofsomething wehaveseenseveraltimes: Exercise 25.32. If2andpare dominant weights, andaisasimple root with ACH) andp(H,) notzero, show thatA+j1~aisadominant weight and T,@T, contains theirreducible representation F.,-. with multiplicity one. So TOP, =F, Olay. others. Incase t=2,withA(H,) #0,Sym?(F) contains F,,, while A(T) contains Tee Exercise 25.3. If44isadominant weight foreach weight ¢of[,show thatthe irreducible representations appearing inT,®Fare exactly theFsInfact,withnoassumptions, everycomponent ofF,@[alwayshasthisform.Onecanshow that Na, isthedimension of {ve(G5:HO) =0,1sinh =HD). Forthis,see(Zel,§131]. Forother general formulas forthemultiplicities N,,, see[Kem], [K-N], Ui} and [Kumt], [Kum]. Wehave also seen aformula fordecomposing therepresentation Iof 426 28.MoreCharacterFormulas GL,whenrestrictedtothesubgroupGL.y_,©.Inthiscasethemultiplicitiesoftheirreducible components againhaveasimplecombinatorial description. ‘There aresimilar formulas forother classical groups Intheliterature, such formulas areoften called “branching formulas," or“modification rules.” We willjust state theanalogues ofthis formula forthesymplecticandorthogonal cases: For80;,€ <803,,,C, and T,theirreducible representation of$03_4,Cpivenby=(A,2°">A20),therestriction is Resize) =Ol (2534) thesumoverallX=(Zy,..-, 2)with MeMekezheo2hiekelhh withthe2,and4,simultaneously allintegersorallhalfintegers. For807.-1€€50740,andItheirreduciblerepresentation of£0;,€given byAmy 2-2 AD, Resta) =ON, (2539 thesumoverall=(2,,..., 41) with, Meh ehezhen2hy 2th withtheJ,and4,simultaneously allintegers orallhalfintegers. For$p3_-2€&sp24C,andI,theirreducible representation ofsp,,€given by A=(A,>-+>4,20),therestriction is ResterSc(G) =DNulis (2536) thesumoverallF=(iy,..-,2y-4) withI,2+++>Jy,20,andthemultiplicityNyyisthenumberofsequences p,,...,p,ofintegerssatisfying ASMAME Me zAWZ MAO and Pr2AL> pave>Pa2Ayr2Pw Asin thecaseofGLC, these formulas areequivalent toidentities among symmetric polynomials. Thereader mayenjoy trying towork them outfrom {hispointofview,cfExercise23.43and[Boe].Alesscomputational approach isgiven in[Zel].‘Aswesawinthecaseofthegenerallineargroup,thesebranchingrulescan ‘beused inductively tocompute thedimensions oftheweight spaces. For example, for50, consider thechain $0,€ >04-46 >804-,63°>$04€. Decomposing @representation successively from onelayer tothenext will finallywriteitasasumofone-dimensional weightspaces,andthedimensioncanbereadofffromthenumberof“partitions” inchains that start with the givenA.Therepresentations canbeconstructed fromthesechains,asdescribed byGelfand andZetlin, ef.(Zel,§10). $253. Tensor Products andRestrictions toSubgroups an Similarly, one can ask forformulas fordecomposing restrictions for other inclusions, such asthe natural embeddings: Sp,,€ ¢SL;,.C, $0,C ¢SL,C, GLC xGLC cGL,,,C, GLC xGL, <GLC, SLC cSpz,€, SLC cSO,,,,€, SL,C ©SO,,C, fomention just afew. Such formulas aredetermined inprinciple bycomputing what happens togenerators oftherepresentation rings, which isnot hard: one need only decompose exterior orsymmetric products ofstandard representations, ef.Exercise 23.31, Afewclosed formulas fordecomposing more general representations canalso befound intheliterature. Westate what happenswhentheirreducible representations ofGL,,Carerestricted totheorthogonalorsymplectic subgroups, referringto[Lit3]fortheproofs: For 0,€ €GLC, withm=2noF2n+1,given=(A2°>Ay29, Resse)=BNali, 537) thesumoverallX=(Z,>+-->J,20),where N=JNaw with Njx, theLittlewood-Richardson coefficient, and thesum over all5=(5,>5;>---)withall5,even. Exercise 23.38. Show that therepresentation Is,2)ofGLiyC restricts tothe direct sum Fay@Na®Moy ‘over O,C. (This decomposition isimportant indifferential geometry: the Riemann~Christoffel tensor hastype (2,2),andtheabove three components ofitsdecomposition aretheconformal curvature tensor,theRiccitensor,andthescalarcurvature,respectively.) Similarly forSp€GLC, Reset) =BNuli. (25.39) thesumoverallA=(2,2"2A,20),where Ma=ENuw NyistheLittkewood-Richardson coefficient, and thesum isover all n=(ty =122My=42***)with each part occurring aneven number of times, Itisperhapsworthpointingoutthethedecomposition oftensorproducts isaspecial case ofthedecomposition ofrestrictions: theexterior tensor product T,611,, oftwoirreducible representations ofGisanirreducible representation ofGxG,andtherestrictionofthistothediagonalembedding, ofGinGxGistheusualtensorproductF®I.‘Therearealsosomegeneralformulas, validwhenever jisasemisimple Lie 48 25.MoreCharacterFormulas subalgebra ofa semisimple Liealgebrag.AssumethattheCartansubalgebra Hisasubalgebraofh,sowehavearestrictionfromh*toh*,andweassume thehalf-spaces determining positive roots arecompatible. We write jifor weights of§,andwewrite je]7tomeanthataweight4ofgrestrictsto SimilarlywriteWWforatypicalelementoftheWeylgroupofj,and7forhalf thesum ofitspositive weights. IfA(resp. 2)isadominant weight forg (resp. @)letNyxdenote themultiplicity withwhich Tappearsintherestriction Of to, ie, Rest) =@iNut Exercise 25.40*. Show that, foranydominant weight4ofgandanyweightj of, nytt)=©Numalt- Brat) =FNant Exercise25.41°(Klimyk). Showthat Nu=T(-I nt ae a a Exercise 25.42. Show that iftheformula ofthepreceding exercise isappliedtothediagonalembedding ofgingxg,thentheRacahformulaofExercise25:3 results Foradditional formulasofasimilarvein,aswellasdiscussions ofhowtheycanbeimplemented onacomputer, thereareseveralarticlesinSTAMJ.Appl Math. 2,1973. Finally, wenote that itispossible, foranysemisimple Liealgebra 9,to make thedirect sum ofallitsirreducible representations into acommutative algebra, generalizing constructions wesawinLectures 15,§17, and§19.Let Trays cessFa,betheirreducible representations corresponding tothefunda-mentalweights0,...,0,Let A’=Sym'(F, O@Fa) This isacommutative graded algebra, thedirect sum ofpieces A= @Sym, @Sym), where a=(a,,....4,) isanmtuple ofnon-negative integers. Then A*is thedirect sum oftheirreducible representation T,whose highest weight is A=Y.qja, andasumJ*ofrepresentations whose highest weight isstrictly smaller. Asbefore, weight considerations show thatJ"=B,J" isanidealin 80 thequotient At=Ql : §25.3. Tensor Products and Restrictions toSubgroups 429 isthedirectsumofalltheirreducible representations, Theproduct TOL, Tine inthisringisoften called Cartan multiplication; note thatthefactthatTy.» ‘occurs once inthetensor product determines such aprojection, butonly up tomultiplication byascalar. ‘Usingideasof§25.1, itispossible togive generators fortheideal J.IfCis theCasimir operator, weknow that Cacts onallrepresentations and is multiplication bytheconstant c,=(A,4)+(2A,p).ou theirreducible represen tation with highest weight 2.Therefore, if4=Svajw,, theendomorphism C—e,1 ofA*vanishes onthefactor F,andoneach oftherepresentations T,oflowerweightpitismultiplication byc,—¢,#0(ef.(25.2)}.Itfollows:that Jt=Image(C —el: At A Exercise 25.43*.WriteC=¥U,Ujasin§25.1.Showthatfor¥,,...,0_vectors:inthefundamental weightspaces,withv,€[,,andJ’;=5°aj,theelement(C~el)(0,-04°...°0,)isthesumoverallpairsj,k,with1<j<k<m,of the terms (Zee.ites)+UilopUo)~2aae)eya)Alle From thisexercise follows atheorem ofKostant: Jisgenerated bythe elements LUO Vibe)+Uso)Uso")—2,Bor forvel, weT,, with aandffundamental roots. Fortheclassical Lie algebras, thisformula canbeused tofindconcrete realizations ofthering. If ‘onewants asimilar ring forasemisimple Liegroup, onehasthesame ring, ‘ofcourse, when thegroup issimply connected; thisleads totheringdescribed inLectures 15and17forSL,C andSp,,€. ForSO,,.C, little change isneeded when misodd, butthere ismore work formeven. Details can befound in wn LECTURE 26 Real LieAlgebras and LieGroups Inthislector wencatehow tocomplete thelaststepinthe proces ovine ate beginning ofPart TI:totake ourknowlege ofthe classification andrepresentation theory ofcomplex algebras andgroups anddeduce thecorresponding statemensia therealcase. Wedothisinthe istsection, givinga listofthe simple classical elLi algebras andsayingafewwordsaboutthecorresponding groupsandtheir(complex)representations. Theexistence ofacompact groupwhoseLiealgebrahasascomplexi-fleation agiven semisimple complex Liealgebra makes 1possible togive another {indeed theorignal) waytoprove theWeyl character formula; wesketch thisn262 Finally, wecanaskin regard (oreal LiegroupsGaquestionanalogoustooneasked fortherepresentations ofiitgroupsin83%:whichofthecomplexrepresentations VofGactually come from real ones. Weanswer this inthemost commonly - countered cases in§26.3. Inthis final lecture, proofs, when weattempt them, are‘severallyonlysketchedandmayrequiremorethantheusualfortitudefromtheede. §26.1: Classification ofrealsimple Liealgebras and groups §26.2: Second proofofWey!’scharacterformula £26.Real,comple,andquaternionl representations §26.1. Classification ofReal Simple LieAlgebras and Groups ‘Having described thesemisimple complex Liealgebras, wenow address the analogous problem forrealLiealgebras. Since thecomplexificationgo@a ofasemisimple realLiealgebra goisasemisimple complex Liealgebra andwehaveclassified those,wearereducedtotheproblemofdescribing thereaforms ofthecomplex semisimple Liealgebras: thatis,foragiven complex Lie algebra g,findingallrealLiealgebrasgowith {264Classiteation ofRealSimpleLieAlgebrasandGroups 4a 0@eC =9. Wesaw many ofthereal forms oftheclassical complex Liegroups andAlgebrasbackinLectures7and8.Inthissectionwewillindicateonewaytoapproach thequestion systematically, butwewillonly include sketches of proots. Togettheidea ofwhat toexpect, letuswork outrealforms ofsty indetail. Todothis, suppose goisany real Liesubalgebra ofsl,C, with %@yC=al,C. Thenatural thing todois(otrytocarry outouranalysis of semisimple Liealgebras fortherealLiealgebra gg:that isfind anelement Hegysuchthatad(H)actssemisimply ongo,decompose feintoeigenspaces,andsoon.Thefirstpartofthispresents noproblem: sincethesubsetofst,C‘ofnon-semisimple matricesisaproperalgebraicsubvariety, itcannotcontaintherealsubspacego€sl,C,0thatwecancertainlyfindasemisimpleH€go, ‘The next thing istoconsider theeigenspaces ofad{H) acting ong,Of ‘course, ad(H) hasoneeigenvalue 0,corresponding totheeigenspace o= R-H spanned byH.Theremaining twoeigenvalues must then sum tozero, Which leaves just(wopossibilities: ()ad(i) haseigenvalues 2and—2,for2anonzero realnumber; multi- plying Hbyarealscalar, wecantake A=2.Inthiscase weobtain a decomposition ofthe vector space gyintoone-dimensional eigenspaces G0=bo® a2Osa Wecanthen choose X€g;and Y'€g-; thestandard argument then shows thatthebracket [X, YJisanonzero multiple ofH,which wemay take tobe Ibyrechoosing Xand ¥.Wethus have thereal form sf, with thebasis 10 or cm)a=(,-)*-(obr(i) (i)ad(t)has eigenvalues iAand—iAforAsomenonzerorealnumber;again,adjustingHbyarealscalarwemaytakeA=I.Inthiscase,ofcourse,there‘renorealeigenvectors fortheactionofad(#)ongo;butwecandecomposefointothedirectsumofhyandthetwo-dimensional subspacegy.corte-spondingtothepairofeigenvalues iand—1.WemaythenchooseabasisB andCfor gy.) with (H,BJ=C and [H,C]=—B. ‘Thecommutator [B,C]willthenbeanonzeromultipleofH,whichwemayluketobeeither Hor—H(wecanmultiply BandCsimultaneously bya ala p,which multiplies thecommutator (B,C]by”) Inthelatter case, we teethatgoisisomorphic tost:R again: these aretherelations wegetifwe {akeasbasisforst;Cthethree vectors cm) ot 10n=(2 3),0-2), omac=(1_°) an 26Real LieAlgebras and LieGroups Finally, ifthe commutator [B,C]=11,wedogetanewexample: ggiinthis case isomorphic tothealgebra sul,=(A: =—Aandtrace(A) =0}©shC, which has asbasis sp0 o _(02) a(0 Saba(n o}™=n 9 Exercise26.1.CarryoutthisanalysisfortherealLiealgebrass0,Rands0,,R. Inparticular, giveanisomorphism ofeach with either s1,R oFst, This completes our analysis ofthereal forms ofsl,C. Inthegeneral case, wecantrytoapply asimilar analysis, and indeed atleast oneaspectgeneralizes: givenarealformgo<gofthecomplexsemisimple Liealgebra8,wecanfind arealsubalgebra ljpcgosuch that hy@€isaCartan sub-algebraofg=go®C;thisiscalledaCartansubalgebra ofgg.Thereisafurthercomplication inthecaseofLiealgebrasofrank 2ormore: the values onbeofarootaeRofgneednotbeeitherallrealorallpurelyimaginary. We, thus, need toconsider theroot spaces gy,Qa,Soa And gay andthe subalgebra they generate, atthesame time. Moreover aswesawintheabove example, whether thevalues oftheroots a€Rofgontherealsubspacehy arereal, purely imaginary, orneither willingeneral depend onthechoiceof bo. Exercise 26.2. Inthecase ofgo=sly cq=slyC, suppose wechoose as Cartan subalgebra hythespace spanned over fbytheelements 2 0 0 0 00 ny~[0-1 0) and wy=[0 01 0 0-1 0-19, ‘Show thatthisisindeed aCartan subalgebra, andfindthedecompositionof 8intoeigenspaces fortheactionofh=bhp®C.Inparticular, findtheroots‘ofgaslinear functions onb,anddescribe thecorresponding decompositionofGo Judgingfromtheseexamples,itisprobablyprudenttoresistthetemptation totrytocarryoutananalysisofreal semisimple Liealgebras viaananalogue,‘ofthedecomposition g=h®(@g,)inthiscase.Rather,inthepresentbook,wewilldotwothings. First, wewillgive thestatement oftheclassificatiog theorem fortherealforms oftheclassical algebeas—that is,wewillHistall thesimple real Liealgebras whose complexifications areclassical al Second, wewill focus ontwo distinguished real forms possessed byany semisimple Liealgebra, thesplit form and thecompact form. These aretwoformsthatyouseemostoften;andtheexistenceofthe latter inparticu will beessential inthefollowing section. £261.Classincation ofRealSimpleLieAlgebrasandGroups 433 Forthefirst, itturns outtobeenough towork outthecomplexifications0®aC=4OiaoftherealLiealgebrasgoweknow.Thelistis: RealLiealgebra ‘Complexification any 4caC sl,C xst,C sl,H=gl,H/R sl,,€ 80),4R p44 sot £0,€ x£0,C spa, opal span spa,€Xsp24€9 She oe) Papelagit 503, ‘Thelasttwointheleft-handcolumnaretheLiealgebrasofthegroupsU,,¢HandU3H ofautomorphisms ofaquaternionic vector space preserving aHermitian formwithsignature(p,q),andaskew-symmetric Hermitianform, respectively. Weshould firstverifythatthealgebras ontherightareindeedthecomplexi-fications ofthoseontheleft.Someareobvious, suchasthecomplexification (ohRe =5,8 O-s1,R =sh,C. ‘Thesamegoesforso,,Randsp,,R.Next, consider thecompleaificaion of ou,={Aeal,C:'7=—A). Toseethat sl,C =su,@isu,,letMe s1,C, andwrite . M=AM—'M)+4(M+‘)=44+4B; thenA€si,iB€sy,andM=$4~1(/2)B.+Thegeneralcaseofsg€lqissimilariftheformisgivenby(x,y)= '80y,thensu,,¢=(A:'4Q =—QA). Writing M€sl,,4€ intheform M=4(M ~ Q-'M-Q)—GM+i0-"47-Q) landusing@='Q=Q"!=Q,oneseesthat Mest, .@i-s,¢.Forthecomplexification ofs{,,C,embed s1,,Cins\,,Cxsi,€byAr-+(A,4). ‘Given anypair (B,C),write BO-KB+C,8+ 0418-6 B+ =HB+CB+C)—i-(QGB +iC,iB+iC). Forthequaternionic Liealgebra, from thedescription ofGL,H wesawin Lecture 7,wehave aH ={AeglagC: AJ=JA}, ae 26,Real LieAlgebras and LieGroups or ‘i M=AM —J-M-J)—1-00M+10-1MJ) toseethat gM @p C=glysC- Exercise26.3.Verifytherestofthelst. The theorem, which also goes back toCartan, isthat thisincludes the completelistofsimplerealLiealgebrasassociated totheclassicalcomplextypes(A,)}-(D,). Infact,thereareanadditional 17simplerealLiealgebrasasso-ciated with thefiveexceptional Liealgebras The proof ofthistheorem is rather long, andwerefer totheliterature (cf.{H-S], (Hel), (Ar]) forit Split Forms and Compact Forms Rather than try10classify ingeneral therealforms qoofasemisimple Lie algebra ,wewould like tofocus here ontwo particular forms thatate possessed byevery semisimple Liealgebra and that arebyfarthemost ‘commonly dealt with inpractice: thesplit form andthecompact form. These represent thetwoextremes ofbehavior ofthedecomposition 9=, 4®(@Pg,) withrespect totherealsubalgebra gy¢g.Tobegin with,thesplit JormofqsaformgosuchthatthereexistsaCartansubalgebrabo&go(that is,asubalgebra whosecomplexifcation )=hy@C<go®C=gisaCartansubalgebra ofq)whoseactiononjghasallrealeigenvalues—i., suchthatalltheroots a€R= h*ofg(with respect totheCartan subalgebra b=yy@C<9)assumeallrealvaluesonthesubspacebo.Inthiscasewehaveadirect sum decomposition Ho=boO(Di) ofgointohoandone-dimensional eigenspacesj, fortheactionofbo(eachwilljustbetheintersection oftherootspaceq,<qwithgo;eachpairjanj_,will generateasubalgebraisomorphictos1,R.Aswewillseemomentarihe thisuniquely characterizes therealform goof9 Bycontrast, inthecompact form alltheroots a€Rc *ofg(withrespect totheCartansubalgebrab=by@Ccg)assumeallpurelyimaginaryvalved ‘onthesubspace fp.Weaccordingly have adirect sum decomposition 90=boOCD) Ofgointobpandtwo-diinensional spaces onwhich bpactsbyrotation (ea|,willjustbetheintersection oftherootspace9,@9-«Withgo);each|,will, Benerate asubalgebra isomorphic tosu.‘Theexistenceofthesplitformofasemisimple complexLiealgebrawikalreadyestablished inLecture21:onewaytoconstructareal—even rations, {261Classification ofRealSimpleLieAlgebrasandGroups 435 —formgoofasemisimple LiealgebragisbystartingwithanygeneratorX,,{fortherootspaceforeachpositivesimpleroota,,completing ittostandard basisX,,,¥,,.and H,=[X,, ¥,,]for thecorresponding s,,=s1,C, andtaking 80tobetherealsubalgebra generated bytheseelements. Choosing awaytowriteeachpositiverootasasumofsimplerootsevendetermined abasis{H,6, X,€ Qu,Y.€9-«}forgo, in(21.20) TheCartan subalgebra jpofgy istherealspanofthese4.Notethatoncebisfixedforg,therealsubalgebra bosuniquely determined asthespan oftheH,forallrootsa.Thealgebrago isdetermined uptoisomorphism; itissometimes calledthenatural realform fg, Note that thisalso demonstrates theuniqueness ofthesplit form: itis theonly realform ggofgthathasaCartansubalgebrahyactingongowith allealeigenvalues, ‘Asforthecompact form ofasemisimpleLiealgebra,itowesmuchofits ‘significance (aswell asitsname) tothelastcondition in Proposition 26.4. Suppose gisanycomplex semisimple Liealgebra andgo<§ areal form of9.LetbgbeaCurtan subalgebra ofG9. =Yo®€thecorre- sponding Cartan subalgebra of9.Thefollowing areequivalent” ()Each root aRE? ofgassumespurelyimaginaryvaluesonYo,and {foreach root athesubalgebra offogenerated bytheintersection I,of (G.4..) with goisisomorphic tosus;Gil)Therestriction 10gooftheKillingformof9isnegativedefinite; ii)TherealLiegroup Gowith Liealgebra goiscompact. Tn(iii),G,canbetakentobetheadjointformofgg.However,atheorem ofWeyl ensures thatthefundamental group ofanysuch G,isfinite, sothe‘ondition isindependent ofthechoiceofGp.Notealsothat,bytheequivalence with(i)and(i),thecondition (i)must beindependent ofthechoice ofCartan bubalgebra fo.Thisisincontrast withthesplitcase, where werequire onlythatthereexistaCartansubalgebra whoseactiononghasalrealeigenvalues;‘aswesaw inthecaseofs!;R,inthesplitcaseadifferenthymayhaveimaginary Eigenvalues ‘Proor. Westart byshowing that thefirstcondition implies thesecond; this villfollow from direct observation. Tobegin with, thevalue oftheKillingformonH€byisvisibly BUH,H)=¥(aH) <0. Next, thesubspaces |,areorthogonal tooneanother with respect toB,soit remains only toverify B(Z, Z)<0forageneral member Z€1,Todothis, KetXand¥begenerators ofg,andg_,<grespectively, chosensoastoform,{opether withtheircommutator H!=(X,Yastandacdbasisfor#C.Bythe analysisofeal forms ofs1,C above, wemay take asgenerators ofthe algebra ‘gencratedby(,theelementsiH,U=X—YandV=iX+i¥.Ifweset Z=aU +bV=(a+ih):X+(—a+i)¥, 436 26,RealLieAlgebrasandLieGroups then we have d(Z) 0ad(Z) =(a+iby?ad(X) 0ad(X) ~(a?+b*)(ad(X)0ad(¥)+ad(¥)©ad(X)) +(a~iby?ad(¥) ad(7. Now, ad(X) ad(X) and ad(¥) ©ad(¥) have notrace, sowecan write trace{ad(Z)0ad(Z))=—2-(a?+b?)-trace(ad(X) 0ad(¥)).(265) Bydirect examination, intherepresentation Sym*V ofslaC, ad(X) ad(Y) actsbymultiplication by(n—A)(n +2—24>OontheA-eigenspace forI,fromwhichwededucethattheright-hand sideof(26.5) isnegative. Next, weshow that thesecond condition implies thethird. This isimme-diate:theadjointformG,istheconnected component oftheidentityofthegroupAut(qo).Inparticular, itisaclosedsubgroupoftheadjointgroupofanditactsfaithfully ontherealvector space dg,presceving thebilinear formB.IfBisnegativedefiniteitfollowsthatGyisaclosedsubgroupofthe‘orthogonal group SO,,8, which iscompact. Finally, ifweknow that Gyiscompact, byaveraging wecan constructa positivedefiniteinnerproductongginvariantundertheactionofGg. Forany Xinfo,ad(X) isrepresented byaskew-symmetric matrix A=(a) withrespect{0anorthonormal basisofgo(cf.(14.23)},$0B(X,X)=Tr(A°A)=Yesaiydye=—Laz,<0.Inparticular,theeigenvaluesofad(X)mustbe Purely imaginary. Therefore a(h,)<iRand=—aforanyroota,from which ()follows. oO We now claim that every semisimple complex Liealgebra has unique ‘compact form. Toseethisweneed analgebraic notion which is,infact,crucialtotheclassification theoremmentioned above:thatofconjugatelinearincoletion,If=99@xCisthecomplexification ofarealLiealgebrago,thereisa map0:99whichtakesx®2tox@7forxegoandz€C;itisconjugatelinear, preserves Liebrackets, and0istheidentity. Therealalgebra goisthefixedsubalgebra ofo,andconversely, givensuchaconjugate linearinvolution@ofacomplexLiealgebragitsfixedalgebra? isarealformofg,Toprovetheclaim, westart with thesplit, ornatural form, asconstructed inLecture 21 andreferred toabove. With abasis forgchosen asinthis construction, ittnothardtoshowthatthereisauniqueLiealgebraautomorphism ¢ofgthattakes each element ofbtoitsnegativeandtakeseachX,toY,(thisfollows fromClaim21.25).Thisautomorphism ¢isacomplexlinearinvolution whichpreserves thereal subalgebra go.This automorphism commutes with the associated conjugate linear 0.The composite op=gaisaconjugate linear involution, from which itfollows thatitsfixed partq,=g°*isanother realformofg,ThishasCartansubalgebra h,=h’*=fo.Wehaveseenthattherestriction oftheKilling form tobyispositive definite. Itfollows thatisrestriction (0hisnegativedefinite,andhencethata,isacompactformofFinally, thisconstruction ofq,from goisreversible, and from thisonecan deduce theuniqueness ofthecompact form. £261.Classification ofRealSimpleLieAlgebrasandGroups 47 ‘Wemay seedirectly from thisconstruction that H=hO Di, where, ~(4,©g-.)"is arealplane with ,@eC= [email protected]{b,, kb. Exercise 26.6,Verily that(A, =H:1<j<n)isabasis forb,(B,=X_— YorC=i-(X,+¥,)}isabasisfor[,,andtheactionisgivenby UpB=PG.and(4),G]=—P'By wherepistheintegera(H,).Inparticular, b,actsbyrotationsontheplanesly Our classical Liealgebras gallcame equipped with anatural real form ao, andwith abasis oftheabove type. These split forms are: ComplexsimpleLiealgebra Splitform share shar seu ert0 spa spk 50,0 8x4 Exerdise 26.7, Foreach ofthese split forms, findthecorresponding compact form g.. Exercise 268. Letgobearealsemisimple Liealgebra.Showthatasubalgebra bofggisaCartansubalgebraifandonlyitisamaximalabeliansubalgebra andtheadjoint action ongoissemisimple. Exercise26.9*.Startingwitharealformgoofgwithassociated conjugation4,showthatonecanalwaysfindacompactforma,ofgsuchthat4(9,)=Beand such that 8=1@r, wheret=by=90%ge,ndp=go-(i°9,).Suchadecomposition iscalleda Cartandecomposition ofgo. Itisunique uptoinner automorphism. Exercise 26.10°, Foranyrealform goof9,given byaconjugation a,show thatthere isaCartan subalgebra bofgthat ispreserved by9,$09.) is 8Cartan subalgebra ofgo. Naturally, thevarious special isomorphisms between complex Liealgebras (H1,€ &s04€ &sp,C, etc)giverisetospecial isomorphisms among their real forms. Forexample, wehave already seen that st Resit, 50; 5p2R while su,250,R=SIH=0H 438 26,RealLieAlgebrasandLieGroups (cf.Exercise 26.1), Similarly, eachoftheremainingthreespecialisomorphisms ‘ofcomplex semisimple Liealgebras gives risetoisomorphisms between their real forms, asfollows: @ 20,6 ah,€ x51,0‘compactforms:sog9tssiXstysplit forms: 503.2 &s1,R xsl, others: 603, 2sIgC, fH &sityxshR, (i)sp.C =0,6compactforms:gH204split forms: sp4R £05, other: ty,yH&504.1. (ii) SC a8060 ‘compact forms: si, &soft split forms: sl4R =05,5 others: sts,» 2$04.23 Sts,y 2USH; slH &s05,, Inaddition, theextra automorphism of#0, coming from trality givesrisetoanisomorphism ufHl%806... Exercise26.11.Verifysomeoftheisomorphisms above.(Ofcourse,inthecase ‘ofcompact and split forms, these areimplied bythecorresponding iso- morphisms ofcomplex Liealgebras, butitisworthwhile toseethem diteclly imany case.) Real Groups Weturn now toproblem ofdescribing thereal Liegroups with these Lie algebras. LetGbetheadjoint form ofthesemisimple complex Liealgebra9 Igoisarealformofg,theassociated conjugate linearinvolution «ofgthat fixes goliltstoaninvolution ¢ofG.(This follows from thefunctorial nature oftheadjoint form, noting that Gisregarded now asareal Liegroup)ThefixedpointsG”ofthisinvolution thenformaclosedsubgroupofG;itsconnected component oftheidentityGpisarealLiegroupwhoseLiealgebra isgo.Giscalled thecomplexificationofGo ‘Wehaveseenin§23.1thatifT=Tyisthelatticeofthoseelementsinon whichallrootstakeintegralvalues,then2niTisthekerneloftheexponentialmappingexp:b-+Gtotheadjointform.IfboisaCartansubalgebra ofgo;T=exp{b) willbecompact precisely when theintersection ofhywith thekernel2niT”isalatticeofmaximalrank.Inthiscase,TwillbeaproductofncopiesofthecircleS!,n=dim),and,sincetheKillingformonbjisnegativedefinite, thecorresponding tealgroup Gawillalso becompact. Such aGowill bbeamaximal compact subgroup ofG. ‘When Gy<Gisa maximal compact subgroup, they have thesame itredve- iblecomplex representations. Indeed, foranycomplex group G’,eachcomplex $261,Classification ofRealSimpleLieAlgebrasandGroups 439 ‘homomorphism fromGtoG'istheextensionofuniquerealhomomorphism from G,toG’.This follows from thecorresponding factforLiealgebras and thefactthat Gpand Ghave thesame fundamental group. This isanother general fact, which implies thefiniteness ofthefundamental gioup ofGo;we ‘omit theproof, noting only that itcanbeseen directly intheclassical cases: Exercise 26.12¢. Prove that x,(Gg)-*=,(G)isanisomorphism foreachofthe dassical adjoint groups. Exercise 26.13*, The special isomorphisms ofreal Liealgebras listed above Biverisetospecial isomorphisms ofrealLiegroups. Can youfindthese? tis another general fact that anycompact (connected) Liegroup isa Quotient (GxG,xxGxTZ, where theG;aresimple compact Liegroups, T=(S!)* isatorus, andZisadiscretesubgroupofthecenter.Inparticular, itsLiealgebraisthedirectsumofasemisimple compactLiealgebraandanabelianLiealgebra.Thisprovidesanotherreasonwhytheclassification ofirreducible representations inthereal‘compact case andthesemisimple complex case areessentially thesame. Representations ofReal LieAlgebras Finally, weshould say aword here about theirreducible representations (always here incomplex vector spacest) ofsimple realLicalgebras. Insome ‘cases these areeasily described interms ofthecomplex case: forexample, the‘reducible representations ofsi,orslyRarethesameasthoseforsl,C,Le.theyaretherestrictions oftheirreducible representations I,~S,C*corre-sponding (opartitions orYoung diagrams 2.This isthesituation ingeneral whenever thecomplexification q~ao@CoftherealLiealgebra gyisstillsimple:therepresentations ofgooncomplexvectorspacesareexactlythefepresentations ofg.ThesituationisslightlydifferentwhenwehaveasimplerealLiealgebrawhosecomplexification isnotsimple:forexample,theireduc- iblerepresentations ofslyC, regarded asarealLiealgebra, areoftheform T.@T,, where [,istheconjugate representation ofT,The situation in several isexpressed inthefollowing Exercise 26.14. Show that ifgqisasimplerealLiealgebrawhosecomplesifica- tiongssimple,itsirreducible representations aretherestrictions of(uniquelydetermined) irreducible representations ofgIfqoistheunderlyingrealalgebra ‘ofasimplecomplexLiealgebra,showthattheirreduciblerepresentations of fhateoftheformV@W,whereVandWare(uniquelydetermined) irreduc-ibletepresentations ofthecomplex Liealgebra. 440 26.RealLieAlgebrasandLieGroups §26.2. Second Proof ofWeyl’s Character Formula Thetitleofthissectionisperhapsinaccurate: whatwewillgivehereiactuallyasketchofthefirstproofoftheWeylcharacter formula.Weyl,inhisoriginalproof,usedwhathecalledthe“unitarian trick,”whichistosayheintroducesthecompact form ofagiven semisimple Liealgebra anduses integrationon thecorresponding compactgroupG.(Thistrickwasalreadydescribed in§9.3,inthecontext ofproving complete reducibility ofrepresentations ofasemi- simple algebra.) Indeed, themain reason forincluding this section (which is,after all, logically unnecessary) istoacquaint thereader with the“classical” treatmentofLiegroupsviatheircompactforms.Thistreatment followsverymuchthe‘same lines a8therepresentation theory offinite groups. Tobegin with, we replacetheaverage(!/\Gl)Yeeof(g)bytheintegralfgf(g)dy,thevolumeelement djchosen tobetranslation invariant andsuch thatfody=1.Uf p:G+ Aut(V) isafinite-dimensional representation, with character (9) =Trace(p(g) thenfep(g)dueHom(¥, V)isidempotent, anditistheprojection ontothe invariant subspace V°.Sofgxy(g) dye=dim(V®). Applied toHom(¥, W)as before, since Xiemy.m =Zviw itfollows that fTowdu=dim(Homel¥, W). SoifVand Wareirreducible, feted{ivew ioJot MN otherwise. Uptonow,everything iscompletely analogous tothecaseoffinitegroups,andisproved inexactly thesame way. Thelastgeneral fact, analogous tothe basic Proposition 2.30, isharder inthecompact case: Peter—Weyl Theorem. The characters ofirreducible representations span &densesubspace ofthespaceofcontinuous class functions. Itis,moreover, thecasethatthecoordinate functions oftheirreducible matrix,representations span adense subspace ofallcontinuous (orL?)functionson G.Fortheproofofthesestatementswereferto[Ad]or[B-tD].Giventhe fundamental rolethat(2.30) played intheanalysis ofrepresentations offiniteg70ups, itisnotsurprising thatthe Peter-Weyl theorem isthecornerstoneof ‘mosttreatments ofcompactgroups,eventhoughithasplayednorolesofat inthis book. ‘Wenow proceed toindicate how theoriginal proof oftheWeyl character £262. Second ProofofWeyt'sCharacterFormula a formula went inthissetting, Inthissection, Gwilldenote afixed compact group, whose Liealgebra gisareal form ofthesemisimple complex Lie algebra [email protected] seen that 8-90 Bh ‘compatible withtheusualdecomposition g¢=he®(9.®9-.)whencom-plexifed. TherealCartan algebra facts byrotations ontheplanes l. Now letT'= exp(h) <G.Asbefore wehave chosen hsothat itcontains thelattice 2niT” which isthekernel oftheexponential map from hetothe simply-connected form ofgc,0T(SYisacompacttorus. Inthis compactcasewecanrealizetheWeylgrouponthegrouplevelagain: Chaim 26.15. N(TY/T= Poor. For each pairofrootsa,~a,wehaveasubalgebra8,&#l3€<ey with acorrespondingsu,<9,ExponentiatinggivesasubgroupSU()<6. Thecement(_)actsbyAd,takingHto—H,Xto¥,and¥toX.It isinN(7),and,withBasintheprecedingsection,(i”5)=exo(5=). Thenaw(}xia)9actsbyreflectioninthehyperplanea!<6,o Note that 8acting on takes thelattice 2nif” toitself, so9Bacts on T= b/2niT byconjugation. Theorem26.16.EveryelementofGisconjugatetoanelementofT:Ageneralelement isconjugate to||such elements ofT. ‘Sketch ofaproof: Note thatGactsbyeftmultiplication ontheleftcosetspace X=G/T. Forany2€G,consider themapf,:X-»XwhichtakesyTto2yT. Theclaim isthat f,must have afixed point, ie,there isaysuch that y"zy€T.Sinceallfarehomotopic,andXiscompact,theLefschetznumber offis thetopological Buler characteristic ofX.Thefrststatement follows from theclaim thatthisEuler characteristics notzero, Thisis agood exercisefortheclassicalgroups;see[BorZ]forageneralproof.Foranotherproofsee Remark 26.20 below. Forthesecond assertion, check first that anyelement that commutes with every element ofTisin T.Take an“irrational” element xinTsothat its muliples aredense in7:Then foranyyeG, yxy" T=>yTy"! =T,andyay"!=xyeT.Thisgivesprecisely[20]conjugates ofxthatareinT. Corollary 26.17. The clas functions onGaretheW-invariant functions onT. 40 26.Real LieAlgebras andLieGroups Suppose Gisarealform ofthecomplex semisimple group Ge,ie.Gisa real analytic closed subgroup ofGe,and theLiealgebra ofGis ge.ThecharactersonGccanbewritten3-n,e2"thesumoverintheweightltie1Kthey areinvariant under theWeyl group. From what wehave seen, they can beidentified with M-invariant functions onthe torus. Let uswork this ‘utforthe classical groups: Case(A,):G=SU(n+1)TheLiealgebras,,,consistsofskew-Hermitanmatrices, b= sg, shgiRt=(imaginary diagonalmatricesoftrace0}, andT=(ding(e™,..., 2%): 9,=0}Inthiscase, theWeyl group20 isthesyinmetric group &,,., represented bypermutation matrices (with one entry 4:1oneach rowandcolumn, other entries 0)modulo T.Letx:T-+5! correspond otheithdiagonalentrye™".SocharactersonTaresymmetie polynomials in2,..-21+, modulo therelation 2y°..."2yey =1Therefor, characters onSU(n +I)aresymmettic polynomials in2y.--s2su1- Case (B,): G=SO(2n +1) consists ofmatrices with n2%? blocks ofthe form (seas?~sin(2n99)sin@2n9})—_cos(2n3) along thediagonal, andone 1inthelower right corner. Again weseethat T=(SY. This time N(T) willhave block permutations tointerchange the blocks,andalsomatriceswithsomeblocks((5)inthesquaresalongthe diagonal, with theother blocks 2x2identity matrices, with a1inthecornertomakethedeterminant positive;thesetake9to9,foreach{whee ablockis(;)ThisagainrealizestheWeylgroupasasemidirectproductofS,and(Z/29.Withzidentifiedwithe®again,weseethatthecharactersarethesymmetricpolynomials inthevariablesz,+2;i.e,incos(2n9,).cos(2nd,) Case(D,):G=SO(2n)hisasintheprecedingcasebutwithnolowercore.Sincewehavenocommertoputa—1in,therecanbeonlyanevennumberofblocksoftheform(;oreflectingthefactthat$Bisasemidirectproduc.of(Z/27and6,Thistimetheinvariantsaresymmetricpolynomial inhe2,+2;',andoneadditional 11,(z, —2("). Case (C,): G~Sp(2n)-b consists ofimaginary diagonal matrices, Tconsis ofdiagonal matrices with entries e2*°. TheWeyl group ingenerated by £262. Second ProofofWeyt'sCharacterFormula 4a permutation matrices and diagonal matrices with entries which are 1'sand ‘quaternionic j's:Disasemidirect productof(Z/2)* and G,.The invariants aresymmetric polynomials inthe2;+2;". ThekeytoWeyl’s analysis istocalculate theintegralofaclassfunction f ‘onGasasuitable integral over thetorus T.Forthis,consider theimap mG/TX TG, r(xT,y=xyx'. Bywhat wesaid earlier, isagenerically finite-sheeted covering, with (20) sheets, Itfollows that 1fn fwrfyntdy [ 18 Jorrer Now x*(f)(xT, y)=fly) since fisaclass function. Tocalculate x*dy, con- sider theinduced map ontangent spaces n= dx:gh xb—>9 Atthepoint(x9;yo)€G/TxT, (xoe™T, yoe'”)H+ Xoe"*yoe%e- xa! ‘Wewant tocalculate 4cget*ygetTe*x2 Nealovers!) Filioe Vor 85Yooolo¥oxa! T's which is Xal8¥0+YoY~Yox)¥0'(¥oya'¥a') =ole+YoYo!—YoYo")¥o' Now yoyy! =ysince yo€Tandyeh. Tocalculate thedeterminant ofx,‘wecanignorethevolume-preserving transformation x9()xq'.Ifweidentify‘awith g/xb,thematrix becomes T—Ad(¥) 0 yy Sothedeterminant ofn,isdet(l—Ad(yo)). Now(a/¥)c=@auandAd(yo)actsase?*) ong,.Hence dering) =T] (26.18) 4s.function onTalone, independent ofthefactor G/T. This gives Weyl’s Integration formula: t ‘dig=— 1=e01)dy. (26.19) fyliteai,OT errno)di (26.19) Remark 26.20. Thesame argument gives another proofofthetheoremthatG iscovered byconjugates ofT:This amounts totheassertion thatthemap a4 26RealLieAlgebrasandLieGroups G/T xT-+6 ofcompact manifolds issurjective Bywhat wesawabove, forageneric point yp€Tthere areexactly [X8|points inx~"(y) andateach‘ofthesetheJacobiandeterminant isthesame(nonzero)number.Itfollowsthatthetopological degreeofthemapxis,sothemapmustbesurjective. Now(1~e?#)(1 —e-#) =(et—o°*)(e"* —6™,soilwesot d=[cme thendet(x,)=AA,AswesawinLemma24.3,=A,,wherepishalfthesum ofthe positive roots and, foranyweight p, A= X(-trenro, Now wecancomplete thesecond proof ofWeyl's character formula: the character oftherepresentation with highest weight isA,49/A,. Sincewesaw in§24.1 thatAys9/A, hashighest weight Aand(seeCorollary 24.6) itsvalve attheidentity ispositive, itsuffices toshow thattheintegral offox2= where z=Ays4/A,- ByWeyl’ integration formula, 1 3! _—=gig|1188=gig|Asveave I,eali,Bi[, 1 at _ayrenmarn. (orem a,aiaft Oem Ee which concludes theproof. §26.3. Real, Complex, and Quaternionic Representations ‘Thefinaltopicwewanttotakeupistheclassification ofirreducible complex‘representations ofsemisimple Liegroupsoralgebrasintothoseofrealquatem- ionic,orcomplextype.Todefineourterms,givenarealsemisimpleLiegroup GqoFitsLiealgebra gyandarepresentation ofGyoFgyonacomplex vectorspaceVwesaythattherepresentation Visreal,orofrealtype,ifitcomesfromarepresentation ofGyofgyonarealvector space Vpbyextensionof scalars(V=Vp@aC);thisisequivalent osayingthatithasaconjugatelinear‘endomorphism whosesquareistheidentity.ILisquaternionic ifitcomesfromaquaternionic representation byrestriction ofscalars, orequivalently iithag‘conjugate linearendomorphism whosesquareisminustheidentity.Finally,wwesaythat therepresentation iscomplex ifitisneither ofthese. (Compare with Theorem 3.37forfinite groups)Havingcompletely classifiedtheirreducible representations oftheclassicalcomplexLiealgebras,andhavingdescribed alltherealformsoftheseLit $263. Real, Complex, andQuaternionic Representations as algebras,wehaveaclear-cutproblem:todeteminethetypeoftherestriction‘ofeachrepresentation toeachrealform.Ratherthantrytoanswerthisinevery case, however, wewill instead mention some oftheideas that allow us toanswer thisquestion, andthen focus onthecases ofthe split forms (wheretheansweriseasy)andthecompactforms(wheretheanswerismoreinterest- ing,andwhere wehave more tools toplay with). Weassume thecomplexifica-tiongofgoissimple,soirreducible representations ofgoarerestrictions of‘unique irreducible representations ofg(cf.(26.14);inparticular,wehavethe classificationofirreduciblerepresentations bydominantweights, Tobegin with, thetensor products oftworeal, ortwoquaternionic, orofpairofcomplexconjugaterepresentations isalwaysreal;andexteriorpowers‘ofrealandquaternionic representations areequally easy toanalyze, asfor finite groups (seeExercise 3.43). Such tensor andexterior powers may notbe inreducible, but thefollowing criterion can often beused todescribe anirreducible component ofhighestweightthatoccursinsidethem Exercise 26.21*. Suppose Wisarepresentation ofasemisimple group Gthat isrealorquaternionic, andsuppose Whasahighest weight 2that occurs withmultiplicity 1,Show thattheirreducible representations T,with highest weight 2hasthesame type asW. Wemay apply this inparticular tothetensor product 1,@T, oftheirreducible representations ofgwithhighestweightsAand1;sincetheirreduc-‘iblerepresentation T;,,, with highest weight4+1appearsonceinthistensor product, wededuce Exercise 2622*,() IT,andT,,areboth realorboth quaternionic, then T+, isreal, (i)IFT;isrealandI,isquaternionic, then T,.,,isquaternionic. (i)If Tyand F,,arecomplex andconjugate, thenI’isreal ‘Thelasttwoexercisesalmostcompletely answerthequestionoftherepre-sentationsofthesplitformsoftheclassicalgroups:wehave Proposition 26.23. Every irreducible representation ofthesplit forms sly.R, H6401.08, P2aR and£0,,,8 oftheclassical Liealgebras isreal ‘Proor. Ineachofthesecases,thestandardrepresentation Visreal,fromwhich IWfollows thattheexterior powers A'Varereal,from which itfollows thatthesymmetricpowersSym"*(\'V) arereal.Now,inthecasesoft,.,RandsP2eR,wwehaveseenthatthehighestweights«,ofthe representations AMVfork=Jyeesgnforma setoffundamental weights:thatis,everyirreducible representa-tion Thashighest weight a,c forsome non-negative integers ay,..., dy Iifollows that Fappears once inthetensor product. Sym"V.@Sym"(A7V) @---@Sym™("V) 446 26,Real LieAlgebras andLieGroups andsoisreal, (Alternatively, Weyl's construction produces realrepresenta- tions when applied toreal vector spaces) The only difference intheorthogonal case isthat some oftheexterior powers A'V ofthestandard representation must bereplaced inthisdescrip- tion bythespin representation(s). That thespin representations arereal follows from theconstruction inLecture 20,cf,Exercise 20.23; theresult in this case then follows asbefore. a ‘The Compact Case ‘WeturnnowtothecompactformsoftheclassicalLiealgebras.Inthiscase,thetheorybehavesverymuchlikethatoffinitegroups,discussedinLecture5.Specifically, any action ofacompactgroupGponacomplexvectorspace Vpreserves anondegenerate Hermitian inner product (obtained, forexample,bychoosingonearbitrarily andaveragingitstranslatesundertheactionofGy}Itfollows that thedual ofVis isomorphic toitsconjugate, sothat Vwillbe either realorquaternionic exactly when itisisomorphic toitsdual V*.(Iatermsofcharacters, thissaysthatthecharacterChar(V)isinvariantundertheautomorphism ofZ[A]whichtakese(n)toe(—1;forgroups,thissaysthecharacter isreal) More precisely, anirreducible representation of& ‘compact group/Lie algebra willbereal(resp. quaternionic) ifand only ifithas ‘aninvariant nondegenerate symmetric (resp. skew-symmetric) bilinear form,Inotherwords,theclasifcation ofanirreducible Visdetermined bywhether V@V=Sym'VONV ‘contains thetrivial representation, and, iso, inwhich factor. Sodetermining which type arepresentation belongs toisavery special case ofthegeneralplethysmproblemofdecomposing suchrepresentations.With thissaid, weconsider inturn thealgebras st,,ugH, ands0,R. LetTybetheirreducible representation ofsl,C with highest weight A=Laven,whereoy=Ly+:°:+Lyi... 1arethefundamental weightsOfs1,C. ThedualofIwillhave highest weight J-a,_,"a, sothatIwillbe realorquaternionic ifandonly ifa,=a,_,forall, Wenow distinguish thee cases: (i)Ifnisodd,thenthesublatticeofweights4=Shayawitha,=a,.,for allfisrely generated bythesums«7,+«0,fori=1,...,(n~ 1/2.Nowa,isthehighestweightoftheexteriorpowerA'V,sothattheirreducible repre:sentation withhighestweightw,+c,,willappear onceinthetensorproduct AV@N'V =(NV) QV), which byExercise 26.21 above isreal. Itfollows that forany weight A=Ya: witha=a,.,for allt,theirreducible representation Tisreal (iia)If'n=2kiseven, then thesublattice ofweights A=Syay-cy with $263. Real, Complex, andQuaternionic Representations an 4,=a,.forallisfreelygeneratedbythesums0%+Wy-ifOri=Ly...4k—Ly together with theweigltt a4.Asbefore, theirreducible representations with highest weight cy+@,, areallreal. Moreover, incase misdivisible by4the representation MVisreal aswell,since AAVadmitsa symmetricbitineat form NV@NV+A4V =C given bywedge product. Itfollows then asbefore that foranyweight 4= Ya: witha,=a,_;foral,theirreducible representation I;isreal. (ib)Incase niscongruent to2mod 4,theanalysis issimilar tothelastcase except thatwedge product gives askew-symmetric bilinear pairing onA‘V.‘Therepresentation \'Visthusquaternionic, anditfollowsthatforanyweight A=Jay,witha,=a,_;forall,theirreduciblerepresentation Tyisrealif 4,iseven,quaternionic ifaisodd.Insum,then,wehave Proposition 26.24. Foranyweight X=Yay-e ofsug,theirreducible repre- sentation Twith highest weight Ais:complex ifa,#4,forany{;real if = a,.,foralliandmisodd,orn=4k,orm=4k+2andday,i8even;andquaternionte ifa,=dy,forall iand n=4k+2.and dya,, tsodd, Next, weconsider thecase ofthecompact form u,H ofsP24C: Tobegin with,wenotethatsincetherestrictionto1Hofthestandardrepresentation ofsp,,€ onV&C?*is quaternionic, theexterior power A*Visrealforkeven andquaternionic forkodd.Sincethehighestweights,ofAVfork=I... {ormasetoffundamental weights,thiscompletely determines thetypeoftheimeducible representations ofu,H:wehave Proposition 26.28. Foranyweight A=3a,-0 oftg, thetrreductble repre- sentation T;,with highest weight 2isreal ifa,iseven foralloddi,andquaternionic ifaisoddforanyoddi. ‘Next, weconsider theoddorthogonal algebras. Part ofthis iseasy: since theresitiction (0s03,,, ofthestandard representation Vof503444 isreal,toreallitsexteriorpowers,anditfollowsthatanyrepresentation of60941 whose highest weight liesinthesublattice ofindex two generated bythehighestweightsoftheseexteriorpowersisrealItremains,then,todescribethetypeofthespinrepresentation; theanswer,whoseverificationweleaveas Exercise26.28below,isthatthespinrepresentation Tof#024,€(that 'stheirreducible representation whosehighestweightisone-halfthehighestweight ofA'V) isrealwhen n=0or3mod 4,andquaternionic ifn=1or 2Lmod 4.This yields Proposition 26.26. Let«bethehighest weight oftherepresentation NVof$03943C.ForanyweightA=ay0r+"°°+Oy-16q-1+Oy0q/20f802y44R,theIrveducible representation V,with highest weight 2tsreal ifaiseven, orfmts a8 26,RealLieAlgebrasandLieGroops congruent to0or3mod4;ifa,isoddandn=1or2mod4,thenTyis‘quaternionic. (Note that, ineach ofthelast twocases, thefact that every representation iselther realorquaternionic follows from theobservation that theWeyl groupactionontheCartansubalgebra h<gincludesmultiplication by~1)Finally,wehavetheevenorthogonal Liealgebras.Asbefore.theexteriorpowers ofthestandard representation Vareallreal, butwenow have two spin representations todeal with, with highest vectors (inthenotation ofLecture19)=(Ley+°**+LyV/2andf=(Ly+o"+Lyng~Ly)/2.Thefirst‘question iswhether these twoaresll-conjugate orconjugate toeach other. Incase1iseven,asinthecaseofthesymplectic andoddorthogonal algebras,theWeyl group action ontheCartan subalgebra contains multiplication by =1(theWeyl group contains theautomorphism of*reversing thesignofanyevennuinberofthebasis elements L,),80 thatIandTywillbeisomorphic totheir duals; ifmisodd, ontheother hand, weseethat I,willhave —fasa weight, sothatI,and[,willbecomplex representations dual toeachother, ‘We consider these cases inturn ()Suppose first that nisodd, and sayAisanyweight, written as Amayy +°°+ynaWq-7 +Aya +040. ay.4ay,therepresentation T,withhighestweight4willnotbeisomorphictoitsdual, andsowillbecomplex. Ontheother hand, T,,appears onceiaT,@Ty=End),andsoisrea;thus,ifa,..=a,therepresentation [willbereal. (i)If,bycontrast,niseventhenallrepresentations ofs0,4ftwillbeeitherrealorquaternionic. Thehalf-spin representations T,andTparerealifn=0,(mod4),quaternionic ifn=2(mod4),afactthatweleaveasExercise2628Itfollowsthat,with2asabove,Twillberealifeithernisdivisibleby4,oFifaya+aisevensifn =2mod4anda,..+a,isodd,Fwillbequaternionie;InSum,then,wehave Proposition 26.27. Therepresentation Tof80348 with highest weight 2;40+°°+yaya+48+40willbecomplexifnisoddanda,4, 4,3itwillbequaterniontc ifn=2mod4anday.+,sodd;anditwilbe real otherwise. Exercise 2628. Verily thestatements made above about thetypes ofthe spinrepresentation T,oftheorthogonal Liealgebras,ie.,thatthespinrepresentation T,of0344,8 isrealwhen n=0or3(mod 4);andquater,nionicifm=1or2(mod4),andthatthehall-spinrepresentations of$0,4R°arerealifn0(mod4)andquaternionic ifn=2(mod 4).Show, infact,thal, theevesCfoalba Co"Cy=COs)areproduct ofoeot copies ofmatrix algebras over R,C,or#4,with Roccurring form=O£1mod8,€occurringform=£2mod8,andHform=3or4modBs £263.Real,Comples,andQuaternionic Representations “9 Exercise 2629. Show that forarepresentation ¥ofacompact group G, 0ifViscomplex jwe?)=41ifVisreal e =I. ifVisquaternionic. Exercise 26.30*. Show that forarepresentation Vofacompact group, the number ofirreducible real components itcontains, minus thenumber of ‘quaternionic representations, ithenumber oftimes thetrivial representation ‘occurs iny#V intherepresentation ring, where yistheAdams operation (6Exercise 2339), APPENDICES Theseappendices containproofsofsomeofthegeneralLiealgebrafactsthat were postponed during thecourse, aswell assome results from algebra and invariant theory which were used particularly inthe“Weyl construction Schur functor” descriptions ofrepresentations. ‘Thefirst appendix isafairly serious excursion inpolynomial algebra. It ‘roves some basic facts about symmetric functions, especially theSchur polynomials, which occur ascharacters ofrepresentationsofGL,orSL,,and. {Bres determinantal formulas forthem interms ofother basic symmetric polynomials. The lastsection ofAppendix Aincludes some new identities Among symmetric polynoinials, which, when thevariables arespecialized, tapress characters ofrepresentations ofSp, and SO,, asdeterminants inthe sharacters ofbasic representations. Appendix Bgives ashort summary ofsome basic multilinear facts about letior and symmetric powers. The first two sections can beused asa ‘eletence fortheconventions and notations wehave followed; the third ‘eaatains 2general discussion ofconstructions such ascontractions, manySpecialcasesofwhichwerediscussedinthemaintext.‘Thenext three appendices conclude ourdiscussion ofthetheory ofLie “agebras, which began inLectures 9,14,and21.Proofs aregiven, bystandard [Bethods, ofthepromised general results onsemisimplicity, thetheorem on fonjugacyofCartansubalgebras, factsabouttheWeylgroup,Ado’stheorem {hatevery Liealgebra hasafaithful representation, and Levi's theorem that‘plitsthemapfromaLiealgebratoitssemisimple quotient.~The lastappendix develops justenough classical invariant theory tofind {Bepolynomial invariants forSL,C, Sp2,C, andSOC. Thiswasthekeyto‘garproofthatWeyl'sconstruction givestheirreducible representations ofthe ‘mplectic and orthogonal groups. APPENDIX A OnSymmetric Functions JA:Basicsymmetricpolynomials andrelationsamongthem$A2: Proof ofthe determinantal identities $43 Other determinantal identities §A.1. Basic Symmetric Polynomials and Relations among Them Thevector space ofhomogeneous symmetric polynomials ofdegree dink ratiables xj, ....%hasseveral important bases, usually indexed bythe partitions A=(4;=A,=++>Ay20)ofdintoatmostkparts,orbyYoung. diagrams with atmost krows (see§4.1). Welistfour ofthese bases, which areallvalid forpolynomials with integer coefficients, orcoefficients inany ‘exnmutative ring. First wehave themonomials inthecomplete symmetric polynomials: Hy=H Hay Hayy wan ‘whereH,isthejthcompletesymmetric polynomial, ie.,thesumofalldistinctmonomnials ofdegree j;equivalently, »12 1AFnw, Nr=xan Forexample, with three variables, Hay =O42 +5), Hay =PAGE tae EUG £% 4s ‘A.OnSyinmetic Funtiont [Next arethemonomial symmetric polynomials: M=DX, (aay thesum over alldistinct permutations a=(#4...) Of(Ayy...y Aas here X= ft ...oxf, For example, Moy =¥08 E48 +2%, Mao =3433 +35) Thethirdarethemonomialsin theelementary symmetrifunctions.Unikethefirsttwo, these areparametrized bypartitions 1ofdinintegers nolarger’thank,ie,k>m,2°"sy20.Theseareexactlythepartitionsthatarconjugatetoapartition ofdinto atmostkparts.(Theconjugatetoapartition, isthepartition whose Young diagram isobtained from that ofbyinter=changingrowsandcolumns.Wedenotetheconjugateof4by1,althoughthe notation Zsalsocommon.) Forsuch set By EgyEyooEn (As) where E,isthejthelementary symmetric polynomial, ic, BmLeyte ty [eae 3Be For example, Fay Oy$a tH) Eea,oy =Xa +X1%3 +2%. ‘The fourth aretheSchur polynomials, which may bethemost important, although they arelessoften met inmodern algebra courses: bgt pl Sie fbep ad) wy where A=[],<)(%;— x)isthediscriminant, and|q,,|denotes thedeter- minant of akxkmatrix, Forexample, SuanXv 8s aX, Sao =stad +d toe teary 2% ‘The first task ofthisappendix istodescribe some relations among these symmetric polynomials, Forexample, oneseesquickly that Suan =Ea =Hi~Hay Sia.0 =Hao) =Ei~Eas Su,0)'Su.o1 =Sas» + S2.0y- These arespecial cases ofthree important formulas involving Schur poly- nomials, which we state next. The first two are known asdeterminontal $A. Basic Symmetric Polynomials andRelations among Thein 455 Jormulas.ThefirstisalsoknownastheJacobi-TrudyidentityFromgeometry,theisttwoaresometimescalledGiambell’sformulas,andthethirdisPier's ‘formula, Theproof willbegiven inthenext section. hawMy ee (as) Note thatif4,,,=": =&=0,thedeterminant ontheright isthe same asthedeterminant oftheupperleftpxpcorner.Thesecondis En Ewer: Exein Eyet Faye 5.=[Eyapl =| . (a6 Where =(J, 4) isthe conjugate pactition toA Thethird “Pieri” formula tells how tomultiply aSchur polynomial S,by ‘basic Schur polynomial Sy=Hy: SiSm=DS (A?) thesumoverallvwhoseYoungdiagramcanbeobtainedfromthatof2byadding atotal ofmboxes totherows, butwith notwo boxes inthesame column,ie,those¥=(r4,...+)with Heh zMehen-SnzA2z0, andSy,=C4,+m=d+m.Forexaniple, theidentity Sia.Say =See+Se.2»+So.t.n+S20 ‘anbeseenfromthepictures PP EP One can usethePieri and determinantal formulas tomultiply any two Schur polynomials, butthere isamore direct formula, which generalizesPer’formula.ThisLittlewood-Richardsonrulegivesacombinatorialformula {orthecoefficientsN,,,intheexpansionofaproductasanearcombination ‘ofSchur polynomials: When kified, weolen omit zeros atthe endofpatton, 20(m)denotes thepartion feo. 0 456 A.OnSymmetricFunetiont SS, =LNawSy- (As) Here2isapartition ofd,apartition ofm,andthesumisoverallpartitions‘of d+m(each with atmost kparts). The Littlewood-Richardson rulesays that Nig, 5thenumber ofways theYoung diagram forAcanbeexpanded tothe Young diagram for vbyastrict p-expansion. Iffe=(ity)... tah ® svexpansion ofaYoungdiagramisobtained byfirstaddingj1,boxes,accord-ingtotheabove description inPieri’s formula, andputting theinteger |in each ofthese j1,boxes; then adding similarly .,boxes with a2,continuing ‘until finally 4boxes areadded with theinteger k.The expansion iscalled srlt if,when theintegers intheboxes arelisted from right toleft,starting with thetoprowandworking down, andonelooks atthefirstfentries inthis list(forany¢between |and4,+---+#4),eachintegerpbetween |andk~1 ‘occurs atleast asmany times asthenext integer p+ Forexample, theequation Sa.n' Siar=Saas+SaraySay#289.2.0 +Sorat Sa. +Sa.2.40 «anbeseenbylistingthestrict(2,expansions oftheVoungdiagramFF] i 1) z Ff Aproof oftheLittlewood-Richardson rule can befound in[Mac, §1.9}; for theother results ofthisappendix wecangetbywithout using it. Formula (A.7), applied inductively, yields Hy=SaySaysSag=DKade (As) where K,,isthenumber ofways onecanfilltheboxes oftheYoung diagram ofwithAyU's,Ay2's,upto2,R's,insuchawaythattheentries ineachrow, arenondecreasing, andthose ineach column arestrictly increasing. Such &tableauiscalledasemistondard tableauon1oftypeA.TheseintegersK,,tt allnon-negative, with Ku=l and K,.=0 ifa>yp, {AQ ie,ifthefirstnonvanishing A,—4,ispositive; inaddition, K,,=0ifhas more nonzero terms thanj.Forexample, ifk=3,(K,,)isgivenbythemateix, {ALLBasiSymmetricPolynomials andRelationsamongThem 4s oo Pf pols]: _—— Theintegers K,,arecalled Kostka numbers. Exercise A.11, Show thatK,,isnonzero ifandonly if Apt age basiytinboot foralli>1. When A=(1,Ie. 1),Kya, .1yiSthenumber ofstandard tableaux onthe diagram ofjt,where astandard tableau isanumbering ofthedboxes ofa Young diagram bytheintegers 1through d,increasing inboth rows and ‘columns, Weneed one more formula involving Schur polynomials, which comes fromanidentityofCauchy.Lety,,...,Ysbeanothersetofindeterminates,andwrite P(x) and P(y) forthesame polynomial Pexpressed interms of variables x,,..., xadJ,....5 Ya,Fespectively. Theformula weneed is 1. A(x)A(y)der|=]2ACIAO)| Ati=l[Jus an ‘Theproofisbyinductiononk.Tocomputethedeterminant,firstsubtract, thefirstrowfromeachoftheotherrows,notingthat iet Foxy Voxy xy TPxay and factor out common factors. Then subtract thefirst column from each of theothercolumns, thistimeusingtheequation ionenhoea Ty Tn Pox Tx tofactoroutcommon factors. Oneisleftwithamatrixwhosefirstrowis(10... 0),andwhose lower right square liastheoriginal entries. Theformula follows byinduction (ef.(Wel, p.202]). a Another form ofCauchy's identity is 4st ‘A.OnSymnmetic Funtions + =rsas, (as)[Joey 508 thesum overall partitions &with atmost kterms, Toprove this, expand the determinantwhosejentryis(L~sy)"=1xy+xy}4Oneers that foranyI>+++>kthecoefficientofy{ty'?-...-yftisthedeterminant|x}! Bysymmetryofthexandyvariableswehave 1 ut «yh| STL (alg, ‘Combining (A.12) with (A.4) gives (A.13). a Expansion oftheleft-handsideof(A.13)gives 1e =Haloyp) =F HiGoMyl(A.15) Tes ns rwn)LAwMAG). (As) Since thepolynomials /1,a8wellastheM,form abasis forthesymmetric polynomials, one can define abilinear form ¢,>onthespace ofhomo- Beneous symmetric polynomials ofdegree dinkvariables, byrequiring that HM, =bap (A169) where 5,,,is1ifA=xand0otherwise.ThebasicfacthereisthattheSchur polynomials formanorthonormal bassforthsparing: Sy5.)=bie (Aly, Inparticular, this implies that thepairing <,>issymmetric. Equation (A.17) iseasily deduced from thepreceding equations, asfollows, Write S,=Yault, =DPM, forsome integer mattices a,,andBy.Then Su5.)=Daa (A) Inorder that FSIS) =FanlildboaM (0) bbeequal to,4,(x)M,(y), which itmust by(A.13) and(A.15), wemust have Yoat=fy - ‘Thisisequivalent totheequation J°,a,;bj_ =54,Which by(A.18) implies (Atn, Because ofthisduality, formula (A.9) isequivalent totheequation 5,2EKyaMy (alg) A. Basi Symmetric Polynomials andRelations among Them 49 ‘This gives another formula forthese Kostka numbers: K,,isthecoefficientofXinSywhereX=xx‘The identities (A.9) and(A.19) forthebasic symmetric polynomials allow ustorelate thecoefficients ofX?inanysymmetric polynomial Pwith thecoefficients expanding Pasalinearcombination oftheSchurpolynomials. Ifisanyhomogeneous symmetric polynomial ofdegree dinkvariables, andAisanypartition ofdintoatmostkparts,definenumbers¥,(P)and«,(P)by ValP) =[Pay (a2) where [P}, denotes thecoefficient ofX*=x}'...:xf inP,and O(P)= (AP, FH tRNA ER2QecrAgs—(A2N) hereA=[]j<,(x;— x).Wewant tocompare these twocollectionsofnumbers, 4s1varies over thepartitions, ‘Thefirstnumbers y(P) arethecoefficients intheexpression P=YvA(PIMs (a2) forPasalinear combination ofthemonomial symmetric polynomials M,.‘Theintegers «,(P)haveasimilarinterpretation intermsofSchurpolynomials: P=Yo,(P)S,. (A.23) ‘Note from thedefinition that thecoefficient ofX'inA°S, is1,and that no. othermonomial withstrictlydecreasing exponents appearsinA-S,;fromthis, formula (A.23) isevident. Inthis terminology wemay rewrite (A.19) and (AS) as Ky=valS,) =(5,J; =coefficient ofX*inS, (A.24) and Kym(Hy)=(8°Haden dr (A25) Lemma A.26. Foranysymmetric polynomial Pofdegree dinkvariables, WP) =EK-0,(P) Proor. We have PvP, =P=TLo,(P5, =E,PPK Ms -¥(5K,s04(P)My andtheresult follows, since theM,areindependent. a ‘Wewant toapply thepreceding discussion when thepolynomial Pisa Product ofsums ofpowers ofthevariables. LetPj=.x{ +:+- +xf,andfor 40 ‘A.OnSymmetic Functions 1=(i;,...,ig),@d-tupleofnon-negative integerswithSai,=d,set PO PPh ‘These Newton orpower sum polynomials form abasis forthesymmetric functions with rational coefficients, butnotwith integer coefficients. Let n,Q) =oP) Equivatently, Po=Fw,(S,. (Am Fortheproof ofFrobenius’s formula inLecture 4weneed aformal lemma about these coefficients «,(i): ‘Lemma A.28. Forpartitions Aandytofd, ' 1itamp yiTiaAAO={otherwise. Proor. WewilluseCauchy’s formula (A.13). Note that el tog([]1—xu)=Lage 2([]tsa")=§Lepe0n 0 1 trma Benner) L SPrer 1 -¥LAr RTy0,0)S,(0)z(95,0). ‘Comparing with (A.13), theconclusion follows a Exercise A29*. Using thepairing <,>of(A.16), thecoeficients «(= @,(P®) canbewritten @,(i) =(S,,P").{@)ShowthattheNewtonpolynomials areorthogonal forthisparing,and PO, POY =isDit. Equivalently, 1pn S.=EDqjee”, where thesum isover allpartitions P=(iy,....14) with Yai, =4,and 2)= Vig! yl (b)Showthat«,()=5),<Si,My)"H,,P>. SA.. Basic Symmetric Polynomials andRelations among Them 46 We should remark that wehave chosen fowrite our formulas for afixed numberkofvariables, sincethatoftensimplifies computations whenkissmall. Itismoreusualtorequirethenumberofvariablestobelarge,atleastaslarge 18thenumbers being parttioned--or inthelimiting ring with aninfinite number ofvariables, cf.Exercise A.32; theformulas forsmailer karethen recovered bysetting thevariables x,=0fori>k.Forexample,ifk>2we haveS3)=Si+Si.ay Which reduces (0$2,=Sy)when k=1“ThenexttwoexercisesgiveformulasforthevalueoftheSchurpolynomials whenthevariablesx,areallsetequalto1;thesenumbersarethedimensions ofthecorresponding representations. Foraformula forS,(1,..., 1)involvinghhooklengthsoftheYoungdiagramof4,seeExercise64. Exercise A.30*. When x,=x‘~', thenumerators in(A.4) arevanderMonde determinants, leading to ; bec HH0 51Po Te Taking thelimit asx+1,onefinds Anatjri i Si, = P44! )(yy = JA By(A.5)and (A.6) wehave also thefollowing twoformulas: 1 Gi - whereSon o) Salle Thyesethwhereye!=oa Stat (.¥ere yaatenDYyg)weretsnoth)=a Exercise A.31*. (a)Show that 5=SKeX% thesum over allmonomials X*=xf'-...-xf+,where,foranyk-tupleaof non-negativeintegers,K,,isthenumberofwaystonumbertheboxesofthe Youngdiagramof1witha,1's,a;2's,...,ak's,withnondecreasing rows {ndstrictlyincreasingcolumns.Inparticular,theright-hand sideisasym-‘metric polynomial, afactwhich isnotobvious from thedefinition.(b)Deducethat5,(1,..,I)isthenumberofways tonumber theboxes of theYoungdiagram ofj1withintegers fromItok,withnondecreasing rows: andstrictly increasing columns (ic,thenumber ofsemistandard tableaux). Exercise A.32*. Theideaofconsidering symmetric polynomials inanarbi- trarily large number ofvariables canbeformalized byworking inthering‘Amlim A(k),where A(k)denotes theringofsymmetric polynomials ink variables. Then 402 A.OnSymmetiieFunctions N=ZHyHageo=BEBpsosBayo isagraded polynomial ring, with 1andE,ofdegreei.Aringhomomorphism 9:A+ Acan bedefined byrequiring HE) =H, forall ()Show that9isaninvolution: J?=9.Equivalently, UH) =E. (i)112"istheconjugate partition to2,show that HS4) =Sy. (ii)AFB,=xf4 xfsthejthpower sum,show that 9) =(-1, (iv) Deduce theformula Bx=D Kuby (¥)Deduceadualformof(A.7} SiS...=SrEn=DSe thesumoverall partitionswhoseYoungdiagrameanbeobtainedfrom that of2byaddingmboxes,withnotwoinanyrow. (i)Show that 1 (-pEen =r po, a Oe pe, a=Zagh Fay where thesums areover alli=(jy,...,i) with Saiy=d, and 2()=fy!112%... igtd", Notethat Leva = ey §A.2. Proofs oftheDeterminantal Identities ‘Toprove theJacobi-Trudi identity (A.5), note theidentities Xf=Bap Bap tee(May, (A33) forany 1<j<k,p 2kAnd forany0.<m<kand p>k, Hyom—EyHyams+Ealpmen+o(PEHyme0.(AS)Bothofthesefollowimmediately fromthedefiningpowerseriesfortheEyandH,Since these tworecursion relations arethesame, there ateuniversalpolynomials A(p,q)inthevariablesE,..,Bysuchthat $A2.Proofsofthe Determinanta Identities 463 Xf=Alp,Dxpt+Alp,xp?++ACP,RD Sf=Alp.DP"+Alp,2s} (PB,(ass) Hye =ACP. DMgames +AUP, Diamn427°+ACD,RYH Foranyintegers A,,..., 4xthis leads tomatrix identities oxi =(AA+ki, De DanoFy=AG ie a3 (Has y-y=(AB +ki Mayor where )p,denotes thekx'kmatrix whose p,qentry isspecified between the parentheses. The relations (A.34):Isoimply: Lemma (A.37). Thematrices (H,_,) and((—1)*°E,.,) arelower-triangular ‘matrices with ’salong thediagonal, andareinversesofeachother. The identities (A.36) therefore combine togive (fy =FlaggDie(=DPEpoeOPDas (A.38) Taking determinants gives (A.5, since thedeterminant ofthematrix inthe middle is1. Exercise A.29*. Prove theidentity byt[]Ca"=Dib thesumoverallk-tuples (m,,...,m,)ofnon-negative integers withm,>1,> m,2°" >m,>|,,anddeduce Pieri’s formula (A.7). Tocomplete theproofs oftheassertions in§A.1, weshow that thetwo determinants appearing intheGiambelli formulas (A.5) and(A.6) areequal, fieA= yy onAy)andf=(Hy--ymy)ateConjugatepartitions,then Wlagyoal =IBash (Ao) HeretheH,andE,canbeanyeleinents (inacommutative ring)satisfying theidentity (St) (5(—1 Et)=1,with Hy=Ey=1andH,= E,=0for <0. Toprove it,weneed acombinatorial characterization oftheconjugacy ofpartitions: Exercise AAI*. For1=(Ay,.0.Ag)andyt=(jt,,....,)Conjugatepartitions, show thatthe sets (tnat-bt<isk} and fntj—ytsisl} formadisjointunionoftheset{1...,k+}. Wealso need abasic matrix identity which relates minors ofamatrix to minors ofitsinverse (ormatrix ofcofactors). IfA=(ay)isanrxrmatrix, and=(5,,....54)and T=(t,...4)aretwosequences ofkdistinctintegers 464 A.OnSymmetricFunctions from {1,...,r}, letAs,r denote thecorresponding minor: As,y isthedeter- rminant ofthekxkmatrix whose ientry isg,,. Lemma A.42. LetAand Bberxrmatrices whose product isascalar matrix: evI,.Let(S,S')and(T,T’)bepermutations ofthesequence(1,...,r),whereS andTconsists ofkintegers, S'andT’ofr~k.Then ON Aap =edet(A) Br where&istheproduct ofthesignsofthetwopermutations. Proor. Bypermuting therows andcolumns ofA,multiplying ontheleftand right bypermutation matrices Pand Qcorsesponding tothetwopermutations(of(y---y7),Wemaytakethe(S,T)minortotheupperleftcorner: Ay AY Then -ippt (31B Now taking determinants intheidentity Ay4)hBY)_(A0 As As) \0BA) \As hens, givestheequation det(PAQ)-det(B,) =det(A,)-c'™*. Since&istheproductof thedeterminants ofPand Q,thelemma follows. i) ProoFoF(A.40).ApplythelemmatoA=(H,.,)andB=((—1)-?E,.,), with r=k+land SHaQ+hAtk betDy SET yk2dykTIh T=(kKk-1,..., 0, TARA QoD. Then Ase =detHaons4-9-wsr-n) =Hagel Similarly, Brg==D Eyiy)=(=DEM—DEEgod =(-fEyersh withd=Sy=4, Since ¢=(—1},(A.40) follows. o §A.3. Other Determinantal Identities 465 §A.3. Other Determinantal Identities Inthisfinal section weprove some variations ofthese formulas which are useful forcalculating characters ofsymplectic and orthogonal groups. We want tocompare minors, notofH=(H,_,) andE=((—1)'"/E,_,), butofmatricesH*andE~constructed fromthembythefollowing procedures: Foran rxrmatrix H=(H,,), andafixed integer kbetween 1andr,H* denotes therxrmatrixobtainedfromHbyfoldingHalongthekthcolumn, ‘and adding each column totheright ofthekthcolumn tothecolumn the same distance tothe lef.That is, tig={lst Haas ith<k UH, ifjak (withtheconvention thatH,,,=Oifporqisnotbetween1andr).Thematrix E~isobtained byfolding Ealong itskthrow, andsubtracting rows above this row from those below: few fu Emus itt>ke ute, tisk Lemma A.43, IfHand Earelower-triangular matrices with W'salong the diagonal, thatareinverse toeach other, then thesame istrueforI1*andE~. Proor.Thisisastraightforward calculation: thei,jentryofthematrixH*-E~ k Fig+MaanadEns+Haba+SHlEp~Ener) =SMasbas+Leae SMa anes Thefirstsumis, andtheothers cancel term byterm. oO Proposition A.44. Let2=(A,,..., Ag)andjx=(41y,..-, 14)beconjugate parti- Hos. Set, Bim6,forist, andjmE,~Biaforiz2Thenthedeterminant ofthek kmatrixwhoseithrowis Isequaltothedeterminant oftheIx.matrixwhoseithrowis Each ofthese determinants tsequaltothedeterminant MBytes ~Boel 466 ‘A.OnSymmetricFunctions ‘and tothe determinant WijHirsh whereHf=H,fori1,andfori>2 . My ViisoddHy=WtMaatMatt {vitew Proor.WithH=(i,.,)andE=((—1)}"7E,.,) wecanapplythebasiclemma(AA2) tothenew mattices A=H1*andB= E~,andthesame permutations (5,8) and (7,77)used intheproof of (A.40).This time Agr =detHisnsranyeah and Ht octet Hapa i= ookHansa =Yip ity=1 Similarly, By =del(ERssay-n) with Exvanzen =(=DMEaay—Eger[Asbefore,LemmaA.42impliesthatthedeterminant ofthefirstdisplayed‘matrix ofthe proposition isequal tothat ofthethird. Noting that Eyres~Byars=Baring+Byysas-a t°°+Eyepear foneeandoelementary columnoperations onthethirdmatrix,subtractingthefirstcolumn from thethid, then thesecond bythefourth, et,toseetha tivesecond andthird determinants areequal. Since If,=117—Hj» thesame argument shows theequality ofthe frst and fourth determinants a Note that inthese four formulas, asin thedeterminantalformulasforSchur polynomials,ifapartitionhaspnonzeroterms,onlytheupperlllpxp subdeterminantneedstobecalculated. WedenotebySythedeterminantof theproposition: SeyMlacinrMayra+HaceosMarieHay-rasak (AAS) Dually setHj=Hy~Hy. and67=Ey+Ey-a +Ba + Corollary AAG. Thefollowing determinants areequal: (ito Hiern +Hig oeMigrant Hiarneah co) WBycisnEgoieatbEgeoeEgctar+Eyt-teak §A3. Other Determinantal Identities 467 ) VBpctty —Barts w) Waotoy —Hagorsh Define5,4tobethedeterminant ofthis corollary: Sam icin Wiper +Miao Mice +Mi-rasah (A47) Exercise A.48*. LetAbethering ofsymmetric polynomials, 9:A+ Athe involution ofExercise A.32. Show that HSeay) =Sn ‘when 2and j:areconjugate partitions. For applications tosymplectic and orthogonal characters weneed to specialize thevariables x,,..., x4. First (for thesymplectic group Sp,,) take k=2n,let2,,..., 2,beindependent variables, andspecialize XUEcoeSatSMetPZToyXaZe Set GmHeyyoesta2sos2") (aay) inthefieldQ(2,,...,2.)ofrational functions. Proposition A.50. Given integers A,>°--2A,20,wehave perth gptieentenypater =ahbpeTHl whereJ,denotes thenxnmatrixwhoseithrowts Crete Syeteat Sacro Sictint Seton) FromProposition A.44weobtainthreeotherformulas fortheright-hand side, 8 Wal=epy-tey —See-tesls (AS) where 6;=E,(2,,--+5 2m274,++25"),andjistheconjugate partition to2. Exercise A.52,Calculate thedenominator oftheleft-hand side: Hep FO AR eee Cyeaeden where&=2)+27!and6=2)—27'. Proor OFProposrtion A.50. Set Gl)=af=35% Ep)=af+3)". (53) 468 A.OnSymmetcic Functions BythesameargumentthatprovedtheJacobi-Trudy formula(A.5)via(A.38,theproposition follows from thefollowing lemma: Lemma AS4, For1<j<nandanyInteger 1>0,((l) isthe product ofthe Lx mm xn, and nxtmatrices co) ia) Ge Sens tbIeoner oeSett+deanerd (=D) 0) Proor. From (A.37) wecancalculate 2}andz;',andsubtracting gives Pa GD=Sodeans oss (ASS) where §,=D2,(— 1%, 2—4).Multiplying (A.33) by27andsub. tracting wefind CAP) ete —We +(=PeSD) == Pept (D+(=DPPepeab2)4+eantfQn— pe(A.56) Note also that (=1e,=(=1p (Asn since D(— fe,” =[](t=2e)( =74)=[]l=Git+P). From (A.56) and (A.37) follows Step Sp Teonsts (Ass) wherery=D-p(—ItFeq-pf(n+1—9):Combining (A.55)and(A.$8)con.cludes theproof. o [Next (fortheoddorthogonal groups Oy,,,) letk=2n+1,andspecializethevariables x,,...,:X2¢8above,andx,,,,'»1.Weintroducevariablesziand2/",squarerootsofthe variables just considered, and wework inthe fietdQ(z!",..., 214), Set Ky=Wyle, osteeton te=Hil D ws) Hess FueBs BN Hyealeay cesta2ceea4 Dy where Histhejthcomplete symmetric polynomial in2n+Ivariables. Proposition A.60.Givenintegers4,2+">220,wehaveIepertiiegetapr rary Ki §A3. Other Determinantal Identities 469 where K,4sthenxnmatrixwhoseithrowis (Rays Karte} Kaen Kaycine t+Kay-iemead Corollary A.46 gives three alternative expressions forthisdeterminant, ¢g., IKAh=Wiserey~haus (Asn wherehy=Hye,00.42mFibseonetyI Exercise A.62.Calculate thedenominator ofthe left-hand side: iia aia ee REARS eee ProorOFPROPOSITION A.60.Wehave(jl)=z}—2;and&(l)=z}+2/"in Q(z, ...,2!)forLaninteger orahalfinteger. First note that EGO =G+ +GD. ‘Multiplying the numerator and denominator ofthe left-hand side of thestatement oftheproposition by€,(4)°...-E(J}, thenumerator becomes {Z,(2, +n—i+1)+(A,+ i)},andthedenominator becomes Kiln =i+1)+Gln—i}=[G(r—E+1).Wecan,therefore,applyLemma ASA tocalculate theratio, getting thedeterminant ofamatrixwhoseentries aresumsofcertain J's.Note thatbydirect calculation K,=Jj+Jj-1, 80the terms can becombined, and theratio isthedeterminant ofthedisplayed matrix K. a Finally (ortheeven orthogonal groups O,,) letk=2n,andspecialize thevariablesx,,...,X29a8above.Set ily oonBunBayonFe boii ) (A683)eyenBeBTeeeBE)MpagneoonBeFeoOD withH,thecomplete symmetric polynomial in2nvariables. Proposition A.64, Given integers A,2+ =Ay20,wehave Iftert4gapteenny{ictit,>0 ee lial if,=0, where L,isthenxnmatrix whose ithrowis (lyetar Latent baer oeLacton tbay-ionea) Asbefore, there areother expressions forthese determinants, €g, Mah=thactsy—Iacecsh (65) wherehy=Hy(24,025Fy)27"000259) 47 ‘A.OnSymmetie Functions Exercise A.66. Calculate the denominator ofthe left-hand side: pt GM =DE asoSab ProoF oFProrosmion A.64. Note that (0)=&(l-+ 1)~&(=1Mult-plyingthenumerator anddenominator by(,°..."fxthenumeratorbecomesIGA,+2—i+1)—GA,+nm—i—Dfandthedenominator becomes je—64)Gln==k=Gln—6+ this isseen bynoting that thebottom row ofthematrix onthelefti Gi) =G(—D)=2G),andperforming rowreductionsstartingfromthe bottomrow.Therestoftheproofisthesameasinthepreceding proposition, The only change iswhen 2,=0,inwhich case thebottom row inthenumerator ‘matrix isthesame asthat inthedenominator a Exercise A.67*. Find asimilar formula for act gettetendhi asiTepe aFl APPENDIX B OnMultilinear Algebra Inthsappendix westate thebasic facts about tEnsor products and exterior and symmetric powers that aeused intheext. Its hoped that areader with some linear algebrabackground canfillindetailsofthe proofs. $8.1: Tensor product 82 Exterior andsymmetric powers §B.3: Duals and contractions §B.1. Tensor Products ‘Thetensor product oftwovector spaces VandWover afield isavector space ¥@ Wequipped with abilinear map VxW+V@W, oxwroOw, which isuniversak foranybilinear map f:VxW-+ Utoa vector space U,thereisauniquelinearmapfromV@WtoUthattakesv®wtofie,w)‘This universal property determines thetensor product uptocanonical iso- morphism, Iftheground field Kneeds tobementioned, thetensor product is denoted V@qW. If{e,}and {fj}arebases forVand W,theelements {e,@fj}forma basis [email protected] can beused toconstruct [email protected] construction is functorial: Hinear maps V—»V’ and W+W"determine alinear map from V@WtoV Ow’. Similarly onehasthetensor product V,®--*@ V,ofnvector spaces, with itsuniversal multiinear map Vx xRKO @ke m B,OnMultilinear Algebra taking 0,x+x0,{00;@**"@ ve(Recall that amap from theCartesianproducttoavectorspaceUismultilinear if,whenallbutoneofthefactors¥;arefixed,theresultingmapfromV,toUislinear.)Theconstruction oftensorProducts iscommutative: VOWRWOV, vOww@e; distributive: MYOWOWXYOWMON®W), and associative: VUONOWzV@VOM=VEVOM, by(U®@wuG(0@[email protected], there aretensor powers V°" =V@-~-@ Vofafixed space V.Byconvention, V®°istheground field IfAisanalgebra over theground field, and Visaright A-module, andW aleftA-module, there isatensor product denoted V@, W,which canbe constructed asthequotient ofV@ Wbythesubspace generated byall (0-0)®w~»@(a:w)forallv¢V,weW,andaeA.Theresultingmapfrom VxWtoV@, Wisuniversal forbilinear maps fifrom VxWtovectorspacesUthatsatisfythepropertythatf(o-a,w)=f(e,a-w).Thistensorproduct isalso distributive. §B.2. Exterior and Symmetric Powers ‘TheexteriorpowersN'VofavectorspaceV,sometimes denotedAlt*V,come‘equippedwithanalternating multilinear map Vix VA, 0,xo OED A A thatisuniversal: forf:Vx==:xV+ Uanalternating multilinearmap,there isauniquelinearmapfromA'VtoUwhichtakesvj4°°"400P04,+04) Recall that amultilinear mapBisalternatingif(oy...v,)=0wheneverwo ‘ofthevectors o,areequal. This implies thatf(e,,-.., 2)changes signwhentwoofthe vectors areinterchanged.! Itfollows that Bieoyys+225Coen)=SEMO)BLC,--o5)foralloeG. ‘Theexteriorpowercanbeconstructed asthequotientspaceofV"bythe subspacegeneratedbyallv,@:-@c,withtwoofthe vectors equal. Welet HVE ARV, m0, BB) =OAA Oy "This follows (rom thestandard polarization: fortwofactors, le+m,04¥)~Bes)= Bos) =oon) Rn) $82. Exterior and Symmettic Powers an denote theprojection. If{e,}iabasis for¥,then feynegavneih<<<h} iabass forA*V. Define APY tobetheground field IFVand Warevector spaces, there isacanonical linear map from AV @AW toMW @W),which takes (0,0°" v,)@ (wy A>Am4) to, A-- 0,Aw A“ AhyThis determines anisomorphism MV@WxDNVON“. Bb (Fromthisisomorphism theassertionaboutbasesofMVfollowsbyinduction‘onthedimension) ‘The symmetric powers Sym*V, sometimes denoted S*V, comes with a universal symmetric multilinear map Vice Vey, opxo eB Be Recallthatamultilinear mapfi:Vx<->xV-+Uissymmetric ifitisunchanged when anytwofactors areinterchanged, or Bless Pay) =BlO 15-01%) forall e©Gy. ‘Thesymmetric powercanbeconstructed asthequotientspaceofV®"bythesubspacegeneratedbyall0,@°""@&,—C9)@"""@CayOFbythoseinwhich @permutes twosuecessive factors. Again Welt VO SymV, (0, OBO) =04-2. denote theprojection. If(,}ibasis frV,then fete ss sh) isabassorSym*V,SoSym*Vcanberegardedasthespaceofhomogeneous polynomials ofdegree ninthevariables e,.Define Sym°V tobetheground field. Asbefore, there arecanonical isomorphisms Sym"(V@1)=@Sym'v@Sym". (2) “TheexteriorpowersA'VandsymmetricpowersSym*Vcanalsoberealized assubspaces ofV®, assuming, aswehave throughout, thattheground field hascharacteristic 0,Wewilldenote theinclusions by,30wehave VO RAV V8, Ve Symi 1S “Theimbedding 1:A'V+V®*isdefined by HoyAoAOa)FE8Bm(e)PayBBPoy (83) (This iswelldefined since theright-hand sideisalternating) The image ofis thespace ofanti-invariants ofthe right action ofG,on¥#" 4 B.OnMultilinear Algebra (Oo) o= 4B" Say HEWES, (BA) (The anti-invariants arethevectors z€V®such that2-6=sgn(o}z forall o€G,) Moreover, ifA=10,then(I/nl)Aistheprojectionontothisanti invariant subspace.’ (Often thecoefficient {/n!isputinfront oftheformula for1;thismakes noessential difference, butleads toawkward formulas for contractions) Similarly wehave 1:Sym"V+Vby Hoy)=FLPatyOBPai es) ‘Theimageof+isthespaceofinvariants oftherightactionofS,onV.If A= 10, then (I/n}/A istheprojection onto thisinvariant subspace. ‘The wedge product determines aproduct MV@NV SAMY, 9 (0,A AOg) Blas AOA ByuadPO, AA OyAgat AOA Dats whichisassociative andskew-commutative. Thisproductiscompatible withtheprojection fromthetensorpowersontotheexteriorpowers,butcaremust‘betakenfortheinclusion ofexteriorintensorpowers,sinceforexamplevAw. issent t0vw ~wo [not to(o@w— wee] by«Ingeneral, the diagram. MV@NY ts “*y | ‘| on ver@yer yomen ‘commutes when thebottom horizontal map isdefined bytheformula (01®rq)@(Omp1B®tea) os) HY 880) 941)@°** @PeayBPoysryB°°BCopmsnyy thesumoverall“shuffles,” ie.,permutations oof{1,...,m+n}thatpreserve theorderofthesubsets{1,...,m}and{m+1,...,1m-+n}. Similarlythesymmetricpowershaveacommutative product(0,..."0,)® (mayose!Btn)#2.Og’Pegsso"OneWithasimilarcompatibility. Note that0?€Sym?Fissentto20@0inV@V,v"€Sym'Vtonl(v@--- @o)in V®, and generally one hastheanalogue of(B.7), changing each “sgn(o)" to “1°informula (B.8). Allthese mappings arecompatible with linear maps ofvector spaces VW, and inparticular commute with theleftactions ofthegeneral linear, group GL(V) =Aut(V) ofautomorphisms, orthe algebra End(V)= Hom(¥, V)ofendomorphisms, onV, AV, andSym*¥. "Tuethisfactorwhichmitsoopreendacosion Lovectorspacesoveridsofcharacterise. $83. Duals and Contractions as Itissometimesconvenienttomakealgebrasoutofthedirectsumofallof thetensor, exterior, orsymmetric powers. The tensor algebra T'V isthe sumzo ¥%, with product determined bythecanonical isomorphism V9"@VE"-«VO""™),TheexterioralgebraNVisthesumBexo/*V,which isthequotient ofTV bythe two-sided ideal generated byallv@o in ¥®.Thesymmetric algebra Sym'V isthesum(yzoSym"V, which isthe ‘quotient ofT"Vbythetwo-sidedidealgeneratedbyall»@w—w@vinV°2. Exercise BY.The algebra Sym'V isacommutative, graded algebra, which satisfies theuniversal property that anylinear map from Vtothefrstgraded Piece C!ofacommutative graded algebra Cdetermines ahomomorphism Sym'V--+C- ofgraded algebras. Use thistoshow that Sym'(V@WV)= Sym'V@Sym'W,anddeducetheisomorphism (B.2).Provetheanalogousassertions forNV, inthecategory ofskew-commutative graded algebras. Inparticular, construct anisomorphism A(V@W)=A'V@NW,where® denotes theskew-commutative tensor product: ititheusual tensor product additively, buttheproduct has(a@b):(¢ @d)=(—1y*#"4*#\(a-b)@(c-d) forhomogencous elements aand cinthefirst algebra, and banddinthe second. Inparticular, thisproves (B.1). §B.3. Duals andContractions ‘Although only afewsimple contractions areused inthe lectures, andmost of these arewritten outbyhand where needed, itmay beuseful toseethegeneral picture, IEV* denotes thedual space to¥,there arecontraction maps gf:VOP@ (VEO YOUN@(VE), forany1<i< pand 1<j<q,determined byevaluating thejthcoordinateofV*)**ontheithcoordinate ofV°*: GO. ®@%,@% BB) (6.10 =Hf001 OBB" Or, @% BBV OB. More generally ifF=(yy .o.5)and J=(jyy.ccy fy) at@two sequences of 1distinct indices from {1,..-,p} and {1,...,g}, respectively, there isa contraction ch:VF@(VA) VOU@(VION (an) whichtakes 010° 41,@ @ ®9,t0 Tl.00.)1©@0,@0,8 @2,@0BOO,BB% For example, if p=q=n and [=J=(1,...,n this contractionVe@(V)%"-+ Cidentifies(V*)®*withthedualspaceofV°" 416 B,OnMultilinear Algebra Now (V®)* consists ofn-multilinear forms onV,and(A"V)* consists of alternating nmultilinear forms onV;inparticular, (A'V)* isasubspace of (V®)*; thisistheinclusion viax*.Thecomposite NV) (VN VON, where thefirstmap istheinclusion 1andthesecond istheisomorphism ofthe preceding paragraph, maps /*(V*) isomorphically onto thesubspace (A’V)*. Explicitly, avy (evye, BEA ABaeDLAADoS,BNO)1/04)+++"Pola) =det(ojo,)1 Thisdualpairing \"'V@\"(V*)— Kisoftendenoted ¢,>.‘There isa similar isomorphism ofSymn"(V*) with Sym’(V)*, butwithout ‘thesigns “sgn(o).” Exercise B.2If¢;,....eqisabasisforV,withefthedualbasisforV*,then {e,AoA eI Sh<< <i,sm} isabasis for AV, and felt-...-el: i,20,Si,=n}isabasis forSym*V, Show that,viatheabove isomorphisms, thedual bases forA"(V*) andSym"(V*) are t lehneemeh}amferia" (eat ‘There arerelated contractions, sometimes called internal products, and denoted sand, onexterior andsymmetric powers. Fortheexterior powers they aremaps: NV @NY(V*) +AUV*) xX@arox ta; ,wv (ay NV ANY +AV), X@arexta, These canbedefined most simply astransposes ofwedge products ie,they aredetermined bytheidentities «axday=(zax,a> forze AV and Kx,p>=(xan B)forpeA(V*). (The relation ofthisdefinition tothecontraction maps cjabove isexpressed inExercise B.16.) Note that when q=0,these contractions reduce tothe previous duality pairing between APYandA(V*),Forsymmetricpowers,theinternalproductsaredefinedsimilarly. SymVE@SmrV Sym", XBAI: ggSym?"*V @Sym*(V*)—Sym4(V), x@arexta - $83. Duals and Contractions an Exercise B.IS. For v,w¢ V,and 9,y€ V*,show that PH AW=VOG— GOW and (0A Wy =elew— gOw)o. More generally, forifx=0A"Avpanda=9)0°"AOpeqWitheV andgeV*,then x40 =Fsen(oeraigsss(O0)--- Poiessn(le) Pay6°APoy thesum over allpermutations ¢of{1,...,P +4} that preserve theorder of (ecg) IP=000°8yyanda=9,AAgythen Gi) Le =sen(o)@(0—)---" Myla) Paypsty AACotpsge thesum over allpermutations that preserve theorder of(p+ 1,....P +4} Verify these formulas andusethem togiveformulas forthese internal productsinermsofstandardbases.Stateandverifyanalogousformulasforsymmetric powers Exercise B.16. Using formula (i)ofthepreceding exercise, show that the contraction map t.maybegivenas1/ptq!timesthecompositionofthemaps NMV @N(V*)-» VOU+O@(V9)PP +VOR AY, where themiddle map isthecontraction map c}of(B.11), with I=J={1,....p},and theothermapscomefrom1andx.Provethesameformulas(with thesame scalar factor) fortheother internal products. ExerciseB.17.Inthesituationofformula(ii),supposethev,areindependent,andletWbethe(p+q}-<dimensional subspace ofVthat they span; supposetheg,areindependent, andletZbethep-codimensional subspaceofVofthecommonzerosofthe @.Show that xa =0ifdim(W -Z) >4,and other-WisexLa=uy°°"4tyforsomevectorsuthatspanWnZ. Exercise B.1B. Prove the formulas (eA sa= xsyse) and xt (erp)=(xLaes. Stateandverifytheanalogous formulasforsymmetric powers.Inparticular,forewe Vg.yeV*, Plo W=Ve +POW and (o-w)L p= wlo)w +p(w)e. Forexample, v-1(97) =29(0)9 and(0°) @=29(0)0. Foradetaileddevelopment oftheseideas,see[Bour,Algebra,Chap3] APPENDIX C OnSemisimplicity §C.1: TheKilling form andCartan’s criterion §C.2: Complete reducibiity andthe Jordan decomposition §C.3: Onderivations §C.L. The Killing Form and Cartan’s Criterion WerecallfirsttheJordandecomposition ofalineartransformation Xofa finite-dimensional complex vector space Vasasum ofitssemisimple and nilpotent parts: X=X,+X,,where X,isthesemisimple part ofX,andX, thenilpotent part.Itisuniquely characterized bythefactthatX,issemisimple(diagonalizable), X,isnilpotent, andX,andX,commute with each other. In fact, X,and X,can bewritten aspolynomials inX,soany endomorphism that commutes with Xautomatically commutes with X,and X,,One case ofthe invarianceofJordandecomposition isaneasycalculation: Exercise C.1*.ForanyX€gi(V),theendomorphism ad(X)ofgi(V)satisfies ad(X), =ad(X,) and ad(X), =ad(X,). ‘There isaKilling form Bydefined ongl(V) bytheformula * BAX, ¥)= TeX oY), (cay where Tristhetrace and©denotes composition oftransformations, Asin (14.23), theidentity By(X,LY,Z))=BALX,¥},Z) cy holdsforallX,¥,Zingl(V). SC.TheKillingFormandCartan'sCriterion 9 ‘The Killing form Bon aLiealgebra qsthat ofExercise C.|fortheadjoint representation: B(X, Y)=B,(ad(X),ad(¥)). This wasintroduced inLecture14,whereafewofisproperties wereproved.HereweusetheKillingformtocharacterize solvability andsemisimplicty oftheLiealgebra Ifgissolvable, byLie’s theorem itsadjoint representation canbeputin ‘upper-triangular form. Itfollows that 2q=[9,9]actsbystrictlyupper- {riangular matrices. SoifXisin2gand Yin,then ad(X)o ad(Y)is strictly ‘upper triangular, inparticularitstraceB(X,Y)iszero.Cartan’scriterionis that thischaracterizes solvability: Proposition C4.TheLiealgebragtssoleabeifandonlyiBlo,2g)=0. Wewillprove something thatlooks alitle weaker, butwillturn outtobe alittle stronger. Weprove: ‘TheoremC5(Cartan’scriterion),Ifgisasubalgebraofgl(V)andB,(X,Y)=0 JorallXand¥ing, then gissolvable. For this, itsuffices toshow that every elementof9isnilpotent,forthen byEngel’stheoremqmustbeanilpotentideal,andthereforegissolvable.Sotake X€9g, and let2y,...y4, beitseigenvalues (counted with multiplicity forXasanendomorphism ofV.Wemust show theA,areall 2e10.These eigenvalues satisfy some obvious relations; forexample, 54,4, = Te(X 0X))=By(X, X)=0.What weneed toshow is RAte+a,=0. (co) Toprove this, take abasis forVsothat XisinJordan canonical form, with2,,..., 4,down thediagonal; thesemisimple partD=X,ofXisthis diagonal transformation. LetDbetheendomorphism ofVgiven bythe diagonal matrix with2,,..., 2,down thediagonal. Since Tr(D0X)=YU,Ay, itsulfices toprove TB 0X)=0. cn Since Xisasumofcommutators [¥,2],with ¥andZing,Te(D0X)is 4sum ofterms oftheform Tr(D °[Y,Z])=Tr({D, ¥] Z).Sowewillbe doneif weknow that[D,¥]belongsto9,forourhypothesis isthatTr(g0g)= O.That is,wearereduced ioshowing ad(D)(9)<6. (ca Forthisitsuffices toprove thatad(B) canbewritten asapolynomial in ad(X), forweknow thatad(X)*(¥) isingifXand¥areing,Since ad(D) = ad(X,) =ad(X), isapolynomial inad(X), isulices toshow that ad(D) canbewiittenasapolynomial inad(D).Thisisasimplecomputation: usingtheusual basis {E} for9l(V), ad(D) andad(D) atecomplex conjugate diagonal tmatrices, and any such arepolynomials ineach other o 480 ©.OnSemisimplicity Wecanprove nowthatifisaLiealgebra forwhich B(99, 2g)=0,thenaissolvable,whichcertainlyimpliesProposition C.4.Bywhatwejustproved,theimage of2gbytheadjoint representation inal(g) issolvable. Since thekerneloftheadjointmapisabelian,thismakes9gsolvable(cf.Exercise98),andbydefinitionthismakes9solvable. Oo Exercise C9. Show that aLie algebra gissolvable ifand only ifBlad(X),ad(X))=0forallXing. ItiseasytodeducefromCartan’scriterionacriterionforsemisimplicity—partofwhichwesawinLecture14,butthereassumingsomefactswehadnot Proved yet Proposition C.10.ALiealgebragissemisimple fandonlyifitsKillingformBis nondegenerate, Proor. By(C.3) thenull-space s={X€g:BUX, ¥)=0forallYe) isan ideal. Suppose gissemisimple. ByCartan’s criterion, theimage ad(s) <gl(@) issolvable; asinthepreceding proof, sisthen solvable, sos=0bythe definition ofsemisimple. Conversely, ifBisnondegenerate, wemust show that any abelian ideal aingmust bezero. IfXea and Yeg, thenA=ad(X)oad(¥)mapsgintoaand«to0,soTr(A)=0.Thisimpliesthat4c6=0,asrequired, a Corollary C.11.Asemisimple LiealgebraisadirectproductofsimpleLiealgebras. Proor.Foranyidealbofg,theannihilator bt=[X92BUX,¥)=OforallYeh)isanideal,by(C.3)again,ByCartan’scriterion,b0!issolvable,hence2er0, 809=®h!. Thedecomposition follows byasimple induction. is) Ifollows that q=9g,andthatallideals andimages ofgaresemisimple, Tnfact: Exercise €.12*, Show that ifisadirect product ofsimple Liealgebras, theonlyidealsingaresumsofsomeofthefactors.Inparticular,thedecomposition into simple factors isunique (not just uptoisomorphism), Exercise C.13*. Show thatifgissemisimple, theadjoint map ad:g->gliq)is anisomorphism ofgontothealgebraDer(g)ofderivations ofg ~ Exercise C.14, Show that ifgisnilpotent then itsKilling form isilentically zeto, andfindacounterexample totheconverse, §C.2. Complete Reducbilty andtheJordan Decomposition 481 §C.2. Complete Reducibility and the Jordan Decomposition ‘Werepeat that thissection isoptional, since theresults can bededuced fromtheexistenceofacompactgroupsuchthatthecomplexification ofitsLiealgebra isagiven semisimple Liealgebra. Weinclude here thestandardalgebraicapproach. Afinite-dimensional representation ofaLiealgebragwill‘becalledag-module, andag-invariant subspaceasubmodule. Proposition C.15,LetVbearepresentation ofthesemisimple LiealgebragandWe Vasubmodule. Then there exists asubmodule W"=Vcomplementary toW. Proof. Since theimage ofgbytherepresentation issemisimple, wemay assume g<gl(V). Wewillrequire aslight generalization oftheCasimir ‘operator C,€End(V) which was used in§25.1 intheproof ofFreudenthal's formula. Wetake abasis U;,..., U,for g,andadual basis Uj,..., U;,butthis time with respect totheKilling form Bydefined inExercise C.1: By(X, ¥)= ‘Tr(X ©¥)(Note byCartan’s criterion that Byisnondegenerate.) Then Cyis defined bytheformula Cy(o) = Uy(Ui-0) AAsbefore, asimple calculation shows that Cyisanendomorphism of thatcommutes with theaction ofgItstrace is THCy)=ETHU,Uj)=FBy(UpUj)=digg) (C16 Wenote also that since Cymaps anysubmodule Wtoitself, and since it ‘commutes with g,itskernel Ker(C,)andimagearesubmodules. Notefirstthatallone-dimensional representations ofasemisimpleqare trivia, since 9gmust acttrivially onaone-dimensional representation, and a= 99, Weproceed {0theproof itself Asshould befamiliar from Lecture 9,thebasiceasetoproveiswhenW<Visanirreducible invariantsubspaceofcodimension one. Then Cymaps Winto itself, andC,actstrivially on¥/W. Butnow bySchur’s lemma, since Wisirreducible, C,ismultiplication bya scalar onW.This scalar isnotzero, or(C.16) would becontradicted. Hence V=W@ Ker(C,), which finishes this special case. Itfollows easily byinduction onthedimension that thesame is(rue whenever Wc Vhascodimension one. For ifWisnot irreducible, letZbea nonzero submodule, andfind acomplementtoW/Z.<V/Z(byinduction},say ¥/Z. Since ¥/Z.is onedimensional, find (byinduction) Usothat ¥=Z® U, Then V=W@U. Bythesame argument, itsuffices toprove thestatement ofthetheorem when Wisirreducible. Consider therestriction map p:Hom(¥,W)-+Hom(Y,1), 482 ©.OnSemisimplicy ahomomorphism ofg-modules. The second contains theone-dimensional submodule Hom,(W, W7).Bythepreceding case, there isaone-dimensional submodule of™'(Hom,(I%, W))<Hom(¥, W)which maps onto Hom,(W, W) byp.Since one-dimensional modules are trivial, this means there is@ srinvariant yin Hom(¥, W)such that p(y) =1.Butthismeans that yisa invariant projection ofVonto W,s0 V=W@Ker(),as required. We will apply this toprove the invariance ofJordan decomposition (Theorem 9.20}. The essential point is: Proposition C.17. Letgheasemisimple Liesubalgebra ofgl(V). Then forany element X€9,thesemisimple part X,andthenilpotent part X,arealso in PROOF. Theidea istowrite qasanintersection ofLiesubalgebras ofgl(V) for which theconclusion ofthetheorem iseasy toprove. Forexample, weknow AcsI(V) since g=g,andclearly X,andX,aretraceless ifXis.Similarly, ifVisnotirreducible, foranysubmodule WofV,let sw=(Ye al(Y): ¥(W) cWandTe(Yly) =0}. ‘Then gisalso asubalgebra ofsy,and X,and X,arealso inSy. Since LX,q]€g,itfollows that [p(X), a]caforany polynomial p(T). Hence [X,, a¢gand[X,, 9]<9,Inother words, X,andX,belong tothe Liesubalgebra nofgl(V) consisting ofthose endomorphisms Asuch that [4,4] <g.Son gives usanother subalgebra towork with. Now weclaim that iistheintersection ofnandallthealgebras syforallsubmodules Wof¥. This claim, aswesaw, willfinish theproof. Letq’betheintersection ofalltheseLiealgebras.Thengisanidealing!sinceg’<n.Bythecompletereducibility theoremwecanfindasubmodule Uofg'sothat g’[email protected] [g,6]€9,wemust have [9,U]=0.Toshow that U is0,itsulfces toshow that forany Y€Uitsrestriction toanyirreduciblesubmodule WofViszero(notingthatYpreservesWsince¥€sw,andthatVisasum ofirreducible submodules). Butsince ¥commutes with g,Schur'semmaimpliesthattherestriction of¥toWismultiplication byascalar,andtheassumption that ¥€6ymeans thatTr(Yly) =0,30 Yly=0,arequired. o Nowifgisasemisimplealgebra,theadjointrepresentation adembedsg ingl(g). ForanyXingthetheorem implies thatthesemisimple andnilpotent parts ofad(X) areing. Wewrite these X,and X,.The decomposition X=X,+X, may becalled theabsolute Jordan decomposition. Note that [XuX_]=0.[follows easly from thedefinition thatifp:g+4’isahomo- morphism from onesemisimple Liealgebra onto another, then p(X,) =p(X), andp(X,) =p(X),. (This follows forexample from thefactthatgisobtained from gbyfactoring outsome ofitssimple ideals) Infact, theabsolute decomposition determines allothers: §C3. OnDerivations 483 CorollaryC.18.1fp:4»gl(V)isany representation ofasemisimple Liealgebra8.thenp(X,) isthesemisimple partofp(X) andp(X,) isthenilpotent partof PAX), Proor. Wejust saw that p(X,) and p(X,) arethesemisimple and nilpotent parts ofp(X) asregarded inthesemisimple Liealgebra g’=p(q). Apply thetheoremtog'<all) fs} IKfollowsthatanelementXinasemisimpleLiealgebrathatissemisimple inonefaithfulrepresentation issemisimple inalrepresentations. §C.3. OnDerivations Inthisinal section wecollect afewfacts relating theKilling form, solvability, andnilpotency with derivations ofLiealgebras, mainly foruseinAppendix E.Wefirstprove acouple oflemmas related totheLie~Engel theory of Lecture 9,Forthese gisanyLiealgebra, r=Rad(a) denotes itsradical, and 99=a, a]. LemmaC19.Foranyrepresentation p:q—»gl(V),everyelementofpga) isanilpotent endomorphism. Proor. Itsuffices totreat thecasewhere therepresentation Visirreducible, forifWwere aproper subrepresentation, wewould know theresult byinductiononthedimension forWand¥/W,whichimpliesitforV.Wemayreplace gbyitsimage, sowemay assume pisinjective. Inthiscase weshow that gx =0,Wemay assume r¥0.Consider thelargest integer ksuch thata=9'tisnotzero.Thisaisanabelianidealofg,ItsufficestoshowthatBq..0=0,forifk>0,thenac9g,We need three facts: (Ifq€ gl(V) isanirreducible representation andbisanyideal ofgthatconsistsofnilpotenttransformations ofV,thenb=0.(Indeed,byEnge!'stheorem, W={veV:X(0)=OforallXeb) isnonzero, andbyLemma 9.13, Wispreserved byg.Since Visirreducible,W=V,whichsaysthatb=0.) {iAtransformation Xisnilpotent exactly when Tr(X*) =0forall positive integers n.(This isseen bywriting XinJordan canonical form.) i)THLX, ¥]-Z) =0whenever [¥,2]=0.(This follows from theidentity (C3: TEX, Y}-Z) =THX-LZD) Next wecan seethat (9,a]=0.For ifX eqand Yea, then [X, Y]€ a; since aisabelian, ¥commutes with [X, ¥]and hence with powers ofLX, YJ. as €.OnSemisimplicity Applying(ii)withZ=[X,YJ"?givesTr(LX,¥)=0forn>0,and(iand (imply that [g,0]=0. Finally weshow that 9go0=0. IfX,Yeq and (X, Yeo, then L.LX, YI]=0bythepreceding step, soagain ¥commutes with powersof [X,Y],andthesameargumentshowsthatTr(LX,YJ")=0,and(i)and() againshowthat2ga=0. fs) Lemma C20. ForanyLiealgebra g,(q,€ inilpotent. Proor. Look attheimages jand ¥ofqandrbytheadjointrepresentation ad:g-»glfq).ByLemmaC.19andEngelstheorem,(G,¥]is@nilpotentideal ofj.Sincethekerneloftheadjointrepresentation isthecenterofg,itfollows that thequotient of[g,r] byacentral ideal isnilpotent, which implies that [4,1] itself isnilpotent. o ‘An ideal @ofaLiealgebragiscalledcharacteristic ifanyderivationofg mapsaintoitself.Notethatanidealisjustasubspacethatispreservedby allinnerderivations Dy=ad(X).Ifollowsfromthedefinitionsthatfaisanyideal ing,then anycharacteristic ideal inaisautomatically anideal in The following simple construction isuseful forturning questions about ‘general derivations into questions about inner derivations. Given any Liealgebragandaderivation Dof,let’=g@C,anddefineabracket ong’by (OG,A,04)=(LX,YJ+AD(Y)~D0,0). Itiseasy toverify that g’isaLiealgebra containing g=9@0asanideal, andthat, setting &=(0,1),therestriction ofD;=ad(€) togisthegiven ‘derivation D. ‘Asa simple application ofthisconstruction, ifBistheKilling form ong, wwehave theidentity BID(X), ¥)+B(X, D(Y)) =0 (2h foranyderivation Dofg,andanyXandYin g,Indeed, ifB'istheKilling formong',(C.3)givesB([Z,XJ,¥)+BX,[é,J)=0;sincegisanidealin4,Bistherestriction ofBTtoq,and(C21) follows. From (C21) itfollows that ifaisacharacteristic ideal ofg,then ils‘orthogonal complement withrespecttotheKillingformisalsoa characteristicideal ofg Proposition C.22. ForanyLiealgebra g,Rad{g) isthe orthogonal complement to9gwith respect totheKilling form. Proor. Toseethatr=Rad(a) iscontained in@g",ie.,that9gisperpendic- ulartox,letX,Yeqand Ze. Recalling that (LX, ¥],Z)=BUX, (%.Z)), itsuffices toshow that B(X, [¥,Z])=0.Lettbethesubalgebraofggenerated byrandX.Then[bb]<r,sohissolvable,sobyLie'stheorem,underthe §C3. OnDerivations 485 adjoint action, hacts ongbyupper-triangular matrices. ByLemma C.19, (YZ) acts ongbynilpotent transformations. Itfollows that X©LY,Z] also acts nilpotently ong,from which itfollows that B(X, LY,Z])= ‘Tr(X 0[¥,Z]) =0,asrequired. Since 9gisacharacteristic ideal, (@q)*isanideal.tissolvablebyCartan’s ctiterion (Proposition C.4), since B(a9!, MBs") <B(a', Bp)=0. Itfollows that %q! cr,which concludes theproof. o Corollary C23. If«isanideal inaLiealgebrag,then Rad(a) =Radig)ne. Proor. Since Rad(a) isacharacteristic ideal ofan ideal itisanideal ofg,Since itissolvable,itmustbecontainedintheradicalofg.Thisshowstheinclusion ©;theopposition inclusion isclear since Rad(q)aisasolvableidealina,a Proposition C.24. IfDisaderivation ofaLiealgebra 9,then D(Rad\() iscontainedinanilpotentidealofq. Proof. Construct =@C asbefore, with €=(0,1). Since Rad(g) < Radi’), wehave D(Rad(q)) =[é,Rad(a)] €[s',Rad(a')} 98. ByLemma C.20, [q’,Rad(q’)] isanilpotent ideal ing,soitsintersection with Gisalso nilpotent a Justaswith thenotion ofsolvability, anyLiealgebra qcontains alargest nilpotent ideal, usually called thenilradical ofg,anddenoted Nil(g) oF1. Proposition C.24 says that anyderivation maps tinto n,which includes the sesult ofLemmaC.20that[9,t]€0.Theexistence ofthisidealfollowsfrom: Lemma C25. Ifaandbarenilpotent ideals inaLiealgebra g,thena+bisalso 4anilpotent ideal. Proor.Anidealaisnilpotentiffthereisapositiveintegerksothatallfold brackets(X,,(Xa.[-.-»Xi-1»Nad.1]arezerowheneachX;isin 4.Equivalently, allm-foldbracketsofmelementsofqarezeroifatleastkofthem are ina.Ifkischosentoworkfor@andfor6,itiseasytoverifythat2k worksforthesum«+6,sinceanybracketof2kelements,eachfromaorfrom b,contains atleast kelements from aorfrom b. a Since Nil(q)<Rad(q),itfollowsfromProposition C.24thatNilisa characteristic ideal ofg.Thesamereasoning asinCorollary C.23gives: 486 ©.OnSemisimplicny Corollary€.26.IfaisanidealinaLiealgebraq,then Nil(a) =Nil(g)4. MisaLiealgebra,itsuniversalenvelopingalyebraU=U(g)isthequotient ofthetensor algebra ofgmodulothetwo-sidedidealgeneratedbyallX@Y~ Y@X~[X, Y]forallX,¥ing,Itisanassociative algebra, with amap 1g» Usuch that ALEX, YD)=D0, YD =201) —HC, and satisfying theuniversal property: forany linear map from gtoan associative algebra Asuch that g({X, ¥})=F9(X), o(7)] forallX,Y,there isaunique homomorphism ofalgebras @:U-+ Asuch that g=01.For cexample,a representation p:4-»l(V) determinesanalgebrahomomorphism B:U(q) -»End(V). Conversely, any representation arises inthis way. Wewill need thefollowing easy lemma: Lemma C21. Foranyderivation DofaLiealgebra ,there is@uniquederivation Boftheassociative algebraU(g)suchthatB1=10D. PRooF. Define anendomorphism ofthetensor algebra ofgwhichiszeroon thezeroth tensor power, and onthenthtensor power is X,@- @Xp DX, @Xz OX,+X,@DAW“ OX,+ +X, @X,@- @DX, ‘Thisiswelldefined,sinceitismultilinear ineachfactor,anditiseasilycheckedtobeaderivation ofthetensoralgebra;denoteitbyD’.ToseethatD”passestothequotient U(@) onechecks routinely thatitvanishes ongenerators for theideal ofrelations. a Exercise C.28. IfDisaninner derivation byanelement Xin,verify thatD istheinner derivation bytheelement (X). Itisafact that thecanonical map 1embeds 9inU(q). The Poincaré Birkhoff- Witt theorem asserts that, infact, ifU(q) isfiltered with thenthpiece generated byallproducts ofatmost nproducts ofelements of1(q),then the associated graded ringisthesymmetric algebra ong.Equivalently, ifXys.--. X,isabasis forg,thenthemonomials X{-...: X'*form abasis forU(q).We 40not need this theorem, but wewil usethefact that these monomialsgenerateUi;thisfollowsbyasimpleinduction, usingtheequations XX;~XX, =LX,Xj]torearrange theorder inproducts. APPENDIX D Cartan Subalgebras $D.t:TheexistenceofCartansubalgebras§D.2Onthestructure ofsemisimple Liealgebras §D3.Theconjugacy ofCartansubalgebras§D4: OntheWeyl group Our task here istoprove thebasic general facts that were stated inLecture14aboutthedecomposition ofasemisimple LiealgebraqintoaCartanalgebra handasum ofroot spaces q,,including theexistence ofsuch and itsuniqueness uptoconjugation. §D.1. The Existence ofCartan Subalgebras Note that ifwehaveadecomposition asinLecture14,andHisanyelement ‘ofhsuch that a(/7) 40forallroots a,then §isdetermined byHH:h=¢(H), where (i)=(Xe9:(H,X)=0}. oH Theelements ofbwith thisproperty arecalled regular. They form aZariski ‘open subset ofh:thecomplement oftheunion ofthehyperplanes defined by theequationsa=0.Inparticular,regularelementsaredenseinb.IfH€his notregular,thencislargerthanbysinceitcontainsotherrootspaces.Note thatallelementsofarealsosemisimple, i.¢,,theyareequaltotheirsemisimple pars. ‘Ofcourse,thisdiscussion depends onknowing thedecomposition which weare trying toprove. Butitsuggests oneway toconstruct andcharacterize 488 D.CartanSubalgebras Cartan subalgebras: they should besubalgebras oftheform ¢(/1) forsome semisimple element 17,that areminimal insome sense. Wecanmeasure this minimality simply bydimension, Definition D.2. The rank nofasemisimpleLiealgebragistheminimumof thedimension of¢(H) asHvaries over allsemisimple elements ofg.A semisimple element Hiscalled regular ifc(H) hasdimension n.ACartan subalgebra ofgisanabelian subalgebra allofwhose elements aresemisimple, and that isnotcontained inany larger such subalgebra. Our first main goal is Proposition D.3. IfHisregular, then e(Il) i@Cartan subalgebra, For anysemisimple element H,qdecomposes into eigenspaces forthe adjointactionofH: 9=Path =®©att, (D4 whereg,(Hf)={X€9:(Hf,X]=AX),and(it)=qo(Hf).Thereisasimilar‘decomposition even ifH(oF4)isnotsemisimple, butreplacing theeigenspace byg,(H1) ={X€g:(ad(H) —AI(X) =0forlarge k}. Exercise D.S. Without assuming that Hissemisimple, show that[as(t.96000]&taeq(llhbyprovingtheidentity (adQety —2+ OMEN, YD. & (k-£(‘)ceaanaN),fadtH)=pty) Letus(temporarily) callanarbitrary elementH€gregularifdim(go(H)) <dim(qo(X)) forallXeg. Lemma D6. IfHisregular, then go(H) isabelian. Proor. Consider how theKilling form Brespects thedecomposition (D.4)— again knowing what toexpect from Lecture 14,If¥isinq,(H) with A#0, then ad(Y) maps each eigenspace toadifferent eigenspace (byExercise D.5), asdoes ad(¥) 0ad(X) forX€go(H}. Thetrace ofsuch anendomorphismis zero, ie. B(X, Y)=0forsuch Xand ¥. Because 9issemisimple, Bisnondegenerate. Sincewehaveshownthat‘A(1)isperpendicular totheotherweightspaces,itfollowsthattherestriction‘ofBtogo(H)isnondegenerate.ConsidertheJordandecomposition X=X,+X,ofanelementXingo(. Sincead(X,) =ad(X), isnilpotent, X,belongs togo(/), 80X,=X—X,doesalso.Thenad(X,)=ad(X),isnilpotent andsemisimple onqo(H),s0itvanishes there. Butthisalready shows that ad(X) =ad(X,) +ad(X,) isanilpotent $02. OntheStructureofSemisimpleLieAlgebras 489 endomorphisin ofgo(Hf)foranyX€go(H4).Hence,byEngel’stheorem,go(H)isnilpotent, sobyLies theorem ghasabasis inwhich theendomorphismsad(X)areupper-triangular forallX©go(H4).Itfollowsthatforanyelementsingol),thetraceofproductsoftheiradjointactionsongisindependent oftheorder ofcomposition. Inparticular, forX,¥,Zego(Hf), thetrace of ad([X, ¥J)¢ad(Z)ongiszero,ie.,B(LX,¥],Z])=0.ButsinceBisnon degenerate ongo(li),[X,¥]=0,80gol)isabelian, o Ifollows immediately that go() isnotcontained inanylarger abelian subalgebra, since any element that commutes with 11ising(I?) bydefini-tion,Tofinishtheproofoftheproposition wemustprovethefollowinglemma, whichalsoshowsthatthetemporary definitionofregularagreeswiththefirst LemmaD.7.IfHisregular,thenanyelementof(1!)issemisimple. Proor. Wesawthat ifisingo(IH) then X,isalso. Using thesame basis as inthepreceding proof, wescethat ad(X,) has astrictly upper-triangular matrix. Hence, B(X,, Y)=Tr(ad(X,)0ad(¥))=0forall¥ingo().Bythe nondegeneracy again, X,=0,asrequired. a Itfollows from Lemma D6that if1isregular, and isingo(H), then (X) contains go(H), andthey areequal exactly when Xisalso regular. Problein D.8*. Prove that ifHis regular inanyLiealgebra, then go({t) isa nilpotent Liealgebra. Exercise D9, Show thatasubalgebra isaCartan subalgebra ifandonly ifit consists entirely ofsemisimple elements and iscontained innolarger sub- algebra with thisproperty §D.2. OntheStructure ofSemisimple LieAlgebras Leth beaCartan subalgebra ofasemisimple Liealgebra g,Under theadjoint representationitconsistsofcommutingsemisimpleendomorphisms. Itsthen standardlinearalgebrafactthatthisactionissimultaneously diagonalizable: a=Bae, (D.10) ‘where theeigenspaces areparametrized bysome setoflinear formsweb*, including «=0,and where a= (Keg: [IX] =a(H)-X forallHeb). Tnparticular, ggisthe centralizer offin, The nonzero aarecalled roots “0 D.CartanSubalgsbras Lemma D.tt.5 =66. Proor. Since fisabelian, liscontained ingoIhcorresponds toaregular clement H,ie, =go) anything that commutes with 17must bein b50 iscontained inb fs Ityis constructed from thetegular clement H,then bydefinition 9,(H)i thedirect sum ofthose q,forwhich a(H) =2.Note that thedecomposition (D.10) may befiner than (D4, butthat if1ischosen tobeanelement of such thatthe a(H) aredistinct fordistinet roots, then thedecompositions coincide (Our next task istostudy theother eigenspaces 9,.ASbefore, wehave[4-99]€fesg:Ifollowsthatif+fi#0,andifXea,andYegythenad(X}o ad(¥)is nilpotent, 80itstrace is2er0, ie, Wfa+#0,thenBlgy.84)=0. x) Now foranyroot, if~awere nota roo, thisimplies 9,isperpendiculartollgincludingf=0),whichwouldcontradictthenondegeneracy ofB.Sowegetone ofthe facts asserted inLecture Id Ifisaroot,then—aisaloaroot (013) Moreover,thepairingB:9,xg-«+Cisnondegenerate Anotherfactaso follows easily The rootsaspany* on) Foriffnot there would beanonzero Xehwith a(X) =Ofor allroots which means that [X,¥]=0forall¥inallg,.Butthen Xisin thecenter of which izero bysemisimplicty ofg Now letabea root, letX€6,Yegaaandtake anyHeb,Then BUH (X, YD)=BUCH, X), ¥)=ACHBUX, Y) 15 ThiscannotbezeroforallH,X,andYwithoutcontradicting whatwehavejustproved. Inparticular, Foranyr00ta8,8-e]#0. 016 LetT,€bbetheelementdualtoviathepairingBomb,ic,characterized bytheidentity B(7, 1)=a(H) foralHin, Weclaim next that [X,Y]=BOX,INT,forallX©4,Ye8 (1) Toseeit,pair both sides with anarbitrary element 11of}. Using (0.15), we have : BUI,BUX,YT.)=BUH,T)B(X, ¥)=a(H1VBX, ¥)=BULLEN. YD. asrequired, Next weshow that 40,3. The Conjugacy ofCartan Subalgebras 41 aT.) #0. (0.18) Suppose thiswere false. Choose X€gy,Yeg-.such thatBUX, Yy= #0.Then[X,¥]=T,,50X,¥,and7,spanaLiesubalgebrasofg.Ifa(7:)=0, sissolvable.Since[X,Ye9s,itfollowsthatad((X,¥1)isanilpotentendomorphisin ofg,Butthen 7,isnilpotent; butallelements ofharesemi- simple, s0T,=0,acontradiction. This gives another claim from Lecture 14: Foranyroota,[es6ata]#0. (D.19) ForwithXandYasabove,[[X,¥],XJ=¢°(T,,X]=c:a(T,)X#0. Thelast remaining factaboutrootspacesleftunproved[romLectureI4is Forany root a,gis one-dimensional (0.2 Bywhatwehaveseen,wecanfindX€g,,Yeg-.,8othat Hf=£X,¥]#0,anda(H)#0.Adjustingbyscalars,theygenerateasubalgebra sisomorphictosizC, with standard basis 1,X,¥,60inparticular a(I¥)=2.Consider theadjoint action ofsonthesumV=§@gue, thesumoverallnonzerocomplexmultipleskaofa.Fromwhatweknowabouttheweightsofrepre-sentations ofs,theonly kthatcanoccur areintegral multiples of. Now sacts trivially onKer(a) €)¢¥,anditactsirreducibly ons¢¥. Together these cover thezero weight space b,since I!isnotinKer(a). Sothe“onlyevenweightsoccurringcanbe0and+2.Inparticular, 2acannot bearoot. (0.21) Butthisimplies that 4acannot bearoot, which says that 1isnotaweight occurring inV.Butthen there canbenoother representations occurring inVie,V=Ker(a)®s,whichproves(D.20) a §D.3. The Conjugacy ofCartan Subalgebras Weshow that anytwoCartan subalgebras areconjugate byaninner auto- morphism oftheadjoint subgroup ofAut(q). FixoneCartan subalgebra b,andconsiderthedecomposition (D.10}.ForanyelementXinarootspace9,, ad(X)¢ alla) isnilpotent, aswehave seen, soitsexponential exp(ad{X)) GLa) isjust afinite polynomial inad(X). Set e(X) =explad(X)) LetE(b) bethesubgroup ofAut(q) generated byallsuch e(X). Wewant to prove nowthatthisgroup isindependentofthechoiceofb,andthatallCartan subalgebras areconjugate byelements inthis group. (We willseeinthenext section that£(b)is theconnected component ofAut(q),i<. that itisthe adjoint group.) Theproof willbeakind ofcomplex algebraic analogue ofthecorte-spondingargumentforcompacttorithatwassketchedinLecture26. 42 D.Cartan Subalgebras ‘TheoremD.22.Lethandlybe0woCartansubalgebras ofg.Then(i)Eth)=E(U}), and(i)there isan element g€E=E(b) sothat g(h) =bi. Proor. FixaCartan subalgebra b,Leta, ..,a,beitsroots. Consider the mapping Pitta, XM, XG defined byF(X, 4XpH)=e(X,)9-9(X,)(H).NotethatFis@poly: nomial mapping from one complex vector space toanother ofthesane dimension, Wewant toshow that notonly isthe image ofFdense, butthat, ieydenotes thesetofregular elements inb,then F(t,%°°"%a,*Deg)ContainsaZariskiopenset,(D.23) ic,itcontains thecomplement ofahypersurface defined byapolynomial equation. Suppose that thisclaim isproved. Itfollows that foranyother Cartansubalgebra Wy,thecorresponding imagealsocontainsaZariskiopenset.Buttwononempty Zariskiopensetsalwaysmeet.InthiscasethismeansE(})Bymeets E(H): Deg.That is,there areg©E(B), H€Dig,g”€Eh), H'HingSUCH thatg(lf) =g'(H"). Butthen since HandHi’areregular, 0)=(90H) =ao(a(#D) =goa’) =9'(ao(')) =9°06 ‘This proves theconjugacy ofland by.And since E()=@E()9"' =E(a16)) =E(a'h) =o'EWV0'" =BW), both statements ofthetheorem areproved. o Toprove (D.23), weuseaspecial case ofavery general factfrom basicalgebraicgeometry:ifF:C+€*isapolynomial mappingwhosederivativeF,|, isinvertible atsome point P,then foranynonempty Zariski open set UCC*, F(U) contains anonempty Zariski open sct.Fortheproof werefer toanybasic algebraic geometry text, eg,[Ha], orto[Bour, VI,App. A}So itsuffices toshow thatdF, issurjective atapoint P=(0,...,0, H),where Hebey. This isasimple calculation: Exercise D.24°. Show that dF,l,(0,...,0,Z)=Z forZeb, and that F,1,(0,-- 50,¥,0,...,0,0) =ad(Y)(H) =—ad(1H)(¥) forYeg,,.ConcludethattheimageofdF],containshandeachrootspace,sodF.|,issurjective.is} Weremark that although thissection, likethepreceding appendix, was written forcomplex Liealgebras, asimple “base change” argument shows: that theresults extend toLiealgebras over anyalgebraically closed fieldofcharacteristic zero.Some,suchasCartan'scriterion,thenfollowoveranyfieldofcharacteristic zero, byextending toanalgebraic closure. . $4. OntheWe! Group os §D4. OntheWeyl Group tnthissection wecomplete theproofs ofsome ofthe general ats about the Weyl group thatwere stated inLectures 14and21.Thenotation willbeasin ‘those sections: Eistherealspace generated bytheroots R;@istheWeyl ‘group, generated bytheinvolutions W,ofEdetermined by WB)=B~Bia=p20, where(,)denotestheKillingform(oranyinnerproductinvariantforthe Weyl group). Weconsider adecomposition R=R UR into positive and negative roots, given bysome |:£+Ras inLecture 14,andweetScR*bethesetofsimple roots forthisdecomposition. Note that for any Win theWes! ero, R= mRyoweRy isthedecomposition into postive and negative root frthetinear mapToW-+,Wewanttoshowthateverydecomposition arisesthisway.Toprove thisweneed some simple variations oftheideas in§21.1 LemmaD25.1/isasimplerotthenW,permealheothepostiverots,ie,W,maps R*\{a} toitself. PROOF. Thisfollows from theexpression ofpositive roots assums f=Y'm,a,. with thea,simple, andthem,non-negative integers. Ifa=a),W,(0) diflers from onlybyanintegralmulloaIa,(ilbaromepostive coeiciens 50 mus bea postive rot it LetWo bethesubgroup ofIgenerated bytheW,,asavaries over the simple roots. (We willsoon seethat IB,=%.) ‘Lemma D.26, Anyroot canbewritten intheform f= W(a) forsome aeS andWeM, Inparteular, R=AS) PRooF. Itsulfices todothisforpositive roots, since YBo(a) =WpW,(a)= —8,(a)for anya eS:1fBis postivebutnotsimple,write=Scmayasabove, andinductontheleve!Ym,Asinthepreviouslemma,thereiasimpleroot 750that(A)[email protected], W,()=W(a)foraeSandWeWy,30ft=W,W(a),as required a Lemma D.27. TheWeyl group isgenerated bythereflections inthesimple roots, let 8, a4 D.Cartan Subalgebras Proor. Given arootfi,wemust show that Wyisin8p, Bythepreceding Jemma, write f=U(@) forsome U©W, aeS,Then Wy=Woy=UW, (02%) sincebothsidesactthesameonandp, a Proposition D.29.TheWeylgroupactssimplytransitively onthesetofdecompositions ofRinto postive and negative roots. Proor. Forthetransitivity, suppose R=Q*UQ”isanother decomposition. Weinduct onthenumber ofroots that areinR*butnotinQ*.Ifthis number iszero, then R*=Q*.Otherwise there must besome simple root « thatisnotinQ*.Itsuffices toprove thatW,(Q*) hasmore roots incommon withR*thanQ*does, forthenbyinduction wecanwrite W,(Q*) =W(R*) forsome We WB,soQ*=W,W(R"), asrequired.Infact,wehavebyLemma bas, WAQ") ORE >WO" ORV {a}=WD" OR {a}, and this proves theassertion. For simple transitivity, wemust show that ifanelement WintheWeyl group takes R*toitself, then itmust betheidentity. Ifnot, write Was@ Product ofreflections insimple roots, We Wy Wes withrminimal,withW,thereflectioninthesimplerootfl.Leta=fj.Itsuffices toshow that WoooWy=WieWhosMoanoeMoy forsomes, |<<r~2.LetU,=Wyay-..:W,-1.Thisequationisequivalent totheequation W,U,W,=U,,orU,W,U;*=W,,ofUa)=f,(sinceby (D.28), Wy =UU"). Tofinish theproofwemustfindanssothatU,(a)=f,-NotethatU,_2(a)= W,_.(a)isapositiveroot(byLemmaD.25,sincef,..#6).Ontheotherhand, thehypothesis implies that Vola)=Wy.Wena)=WiWi(—a)=—Wea) isanegativeroot.SotheremustbesomeswithI<s<r—2suchthatUsa) ispositive andU,.,(a) isnegative. This means that W,takes thepositive root U,Ga){o thenegative root U,(2).ButbyLemma D.25 again, thiscanhappen only iW,is thereflection intheroot U,(a), ie,f,=Usa). a ‘The simple roots$foradecomposition R=R*UR”arecalledabasiTor the roots. SinceSandR*determineeachother,thepropositionisequivalent totheassertionthattheWeylgroupactssimplytransitively onthesetofbases ExerciseD.30,ForW'©2,setIW)=#(R? W(R")). ShowthatWcanbewrittenasaproductofIW)relections insimpleroots,butnofewer. §D4. OntheWeyl Group 495 IF, denotes thehyperplane inEperpendicular totheroota,the(closed)Weylchambersaretheclosuresoftheconnected components ofthecomple-ment €\()Q, ofthese hyperplanes. Foradecomposition R=R*UR-with simple roots S,theset W=(Bek:(P,a)20,Yae R*}=(feE:(P,a)>0,Vaes) isoneofthese Weyl chambers. Thefactthat every Weyl chamber arises this, way follows from Lemma D331. For any in€there issome WeM such that (W(f), 2)>0 forall ae5, PRoOF. Letpbehalfthesum ofthepositive roots. Itfollows from Lemma D.25 that W.(p) =p—«for anysimple root a.Take Win9Btomaximize the inner product (17(B), p).Then forallaeS, (HWP), p)=WB), Wap) =(WCB), p—2)=(WCB), #)—(UB. 2) cannotbelargerthan(H(A),ps0(W(f),a)<0. a ‘Thus,theorbitofoneWeylchamberbytheWeylgroupcoversE,s0all‘Weyl chambers areconjugate toeach other bytheaction oftheWeyl group. Soall arise bypartitioning Rintopositive andnegative roots. This partition- ingisuniquelydetermined bytheWeylchainber.Infact,thewallsofaWeyl‘chamber arethehyperplanes 2,asavaries over thencorresponding simple roots, n=dim(E). From theproposition wehave: Corollary D.32. The Weyl group acts simply transitively onWeyl chambers Exercise D.33*. Let6bethegroup ofautomorphisms ofEthat map Rto itself ()Show that$Bisanormalsubgroupof© (i)Let3%betheautomorphisms in6which map agiven setofsimple roots 5toitself, Show that isasemidirect product ofWand 3.(i,ShowthatBisisomorphic tothegroupofautomorphisms oftheDynkindiagram, (iv)Compute 8foreach ofthesimple groups Our next goal istoshow that thelattice Z(H,:a€ R}<bhasabasis of elements H,where avaries over thesimple roots. This isanalogous tothe statement wehave proved thattheroot lattice A,in*isgenerated bysimple roots. The firststatement canbededuced from thesecond, using theKillingformtomapbytob*,Hy+(H,—),where(,istheKillingform,WesawinLecture {4that thismap takes H,toa’=(2/(e,a))a. Given aroot system R inaEuclidean space E,toeach root aonecandefine itscoroot a’inFbythe formula 496 D.CartanSubalgebras ate fea) Let R'm {a':aeR) bethesetofcoroots. For any O#aeb, seta’= (2ila,a))a, andforanyale, setMy=2(0,ata,a).LetR=R*UR”be decomposition ofRinto positive and negative roots, and letSbethe corresponding setofsimple positive roots. Lemma D.34.(i) ThesetR’ofcoroots forms aroot system in€ (i)Theset$'=a’:eeS)isasetofsimple roots forR. Gi)ForaPES. type=Map. ProoF.Itisa straightforward calculationthatny:=nap.Itfollowsbyanother shortcalculationthatiW,denotesthereflectioninthehyperplanepetpendic- ulartoa,thenW,(f')=(W,((B)¥.Thefourdefiningproperties ofarootsystemspecified in§21.1 follow immediately from this. Itisclear thatifR*isthe setofroots inRthat arepositive forafunctional !onE,then (R*Y = {a’:aeR*)isthecorresponding setofpositive roots forR’.Roots inRare those that canbewritten asanonnegative linear combinations ofroots in S,and thisproperty characterizes S.Since aisapositive multiple of«forany 4,itfollows that roots in(R*) arethose that canbewritten asnon-negative linear combinations ofroots in, which proves (i) a ‘The root system R’icalled thedualofR. ExerciseD.35.Findthedualofeachtypeofsimplerootsystem. Proposition D.36.(i)TheelementsH,fora€SgeneratethelatticeZ{H,:2R). Gi)Ifo, €lyaredefined bytheproperty that«,(H,) =5,then theelements 1,generate theweight lattice Ay. (ii)Thenonnegative integral linear combinationsofthefundamental weights ‘©,areprecisely theweights inW'- Ay, where Wistheclosed Weyl chamber corresponding toR*. ProoF. The isomorphism f)-+* given bytheKilling form takes Hf,tothecoroota’.Bythe lemmaandthefactthatallpositiverootsaresumsofsimple roots, theset{a's cS}spans thesame lattice as(a’:a€R).Thisproves(i, and itfollows that theweights areprecisely those elements inbthat take integral values ontheset(H,:€ S).Therestoftheproposition follows, noting that W=(BEE:PH.)>Oforallae R*}=(Pek:pli)2Oforallae} is) Ifweidentifybwith*bymeansoftheKillingform,wecanregard28as agroup ofautomorphisms ofb,Bymeans ofthis, thereflection W,corre- §D4. OntheWeyl Group 47 spondingtoaroot«becomestheautomorphism ofhwhichtakesanclementHo H—a(H)-H,, Wehave alastdebt (Fact 14.11) topayabout theWeyl group: Proposition D.37. Every element oftheWeyl group isinduced byanauto- ‘morphism of«which maps hoitsel/ Proor. Itsuffices toproduce thegenerating involutions W,inthis way. The claim isthat ifX,and Y,aregenerators ofgyandgu.a8usual, then 9,= e(XvJe(—YJe(X,) issuch anautomorphism, where, asinthepreceding section, wewrite e(X) forexpad(X)). We inust show that 9,(H) =H— a(H)-H, forallHin b,Itsulfices todothisforHwith a(H) =0,and forH=H,,sincesuchtogetherspanb.Ifa(H)=0,then(X,,H]=(¥,,H]=0,508,(If)=I,whichtakescareofthiscase. For H=I,itsulficestocalculate onthesubalgebras, =C(I,Xx,%e}&sl3C,andthisisasimplecalculation: Exercise D.38, (a)For slyC with itsstandard basis, show that 9= e(X)e(V)e(X) maps Hto—H, Xto~¥,and ¥to—X. (b)Show thatifGisaLiegroup with Liealgebra g,then 9,isinduced by theelement exp(4n(X, —¥,))ofG. Weneed arefinement ofthepreceding calculation. Foraroot and anonzerocomplexnumbert,definetwoautomorphisms ofg: B40)=e(°X,)2(OEY)0e(0-Xy) and (0) =9,(0)0 9(—1). Lemma D.39. Theautomorphism ®,(t) istheidentity omb,andforanyrootBi itismultiplication by0%ongy Proor. Look first insty,with X=X,,Y= ¥,.Its simplest tocalculate inthecoveringSLyCofthe adjoint group. Here 9,(0 lifts(0 » -yyepanae(! )(1.(4# expeexyexp(—'v-expO)=(4 1) (ir a) (o 4 °‘) —rt o} $0.() litsto °‘).0-1-('°) -et ofrof~loet}: Tossce how,(0) actsongp,forf#+4,itsullices toconsider theaction oftheSL,Ccorresponding to5,=C{H,,Xq,¥,}ontheastringthroughfii. 498 D.CartanSubalyebras ‘on@apae- Weknow thatthisisanirreducible representation ofSLC, and theweightofg,isf(HH,).Itfollowsthat(53)actsbymultiplication by «ft, Simiarly ontitactsbymultiplication by&°=1. ao Putting thepreceding results together, wecangive adescription oftheautomorphism groupAut(q)ofg.LetE=£(b)bethesubgroupgeneratedbyelements exp(ad(Z)), a8Zvaries over root spaces gy, «#0, asin§D3LetGbetheadjointformofg,sowehave EEG Aut) <Auto, where Aut%(q) istheconnected component oftheidentity. Proposition D0.WehaveE=G=Aut®(q)andAut(g\/Aut®(q isisomorphictotheautomorphism groupoftheDynkindiagram. Proor. FixtheCartan algebra handpositive roots R*.LetAut(gy' bethe group ofautomorphisms ofgthat map 6toitself, andsimilarly denote by primes theintersections ofsubgroups with Aut(q. Weleave ittothereadertoconstructafinitesubgroupKofAvt(qywhichmapsisomorphically ontotheautomorphism group oftheDynkin diagram, and which meets Gonly in theidentity clement (see Exercise 22.25 foradirect case-by-case approach, oFuse(21.28). Itthen suffices toprove that Aut) isasemidirect product ofEandK,ie,thatAut(g)=E°K. ‘Toseethis, start with any element «inAut). ByTheorem 022, thereis at,€Ewitho(b)=14(6).Thena=tj"isinAut(ay.ByPropositionD.29 andtheproofofPropositionD.37thereist,€E’s0thata,="+0,maps R*(oR.Thiselementmaypermutethesimpleroots,butthereissomekeK s0that6,=03k"istheidentityonthesetofsimple roots. Now aisthe identity onhanditismultiplication bysome nonzero scalar cyoneach gy. Bythenonsingularity oftheCartan matrix there issome nonzero complex number ¢andsome 2.¢Agsothatcy=("for every simple rootp.From Lemma D.39 itfollows that there is&rinEso that and ayagree oneach 4,foreach simple root f,andboth atetheidentity on6.Butitthen follows fom theuniqueness theorem (Claim 21.25) thato,=s,Hence omttaykeEK, asrequired. a Exercise D.AL. Show that any two Borel subalgebras ofasemisimple Lie algebra areconjugate. APPENDIX E Ado’s and Levi’s Theorems SE: Levis theorem $62: Adis theorem §E.1. Levi's Theorem ‘Theobject ofthissection istoprove Levi's theorem: ‘Theorem E.1.LetqbeaLiealgebra with radical t.Then there isasubalgebra Nofqsuchthatg=+L. Proor. There areseveral simple reductions. First, wemay assume there isno nonzero ideal ofgthat isproperly contained inr.Forifawere such an ideal, byinduction onthedimension of9,g/awould have asubalgebra complementary (or/a,and thissubalgebra hastheform (/a,with (asrequired. Inparticular, wemayassumerisabelian, sinceotherwise Mrisaproperideal intwhich isanideal inqbyCorollary C.23. Wemay also assume that {9,1] =x,forif a,x]=0thentheadjoint representation factors through 9/t, andsinceg/tissemisimple, thesubmodule t<ghasacomplement, whichis therequired 1 Now V=gl(g)is ag-module viatheadjoint representation: forX€gand vey, X-9=[adX),9]=adX)0.9—poadlX). Inother words, forX,Yegandge¥, (4-0) =EX,@09] =(LX 1D. (2) 500 E.Ado'sandLevi'sTheorems ‘ThetrickistoconsiderthefollowingsubspacesofV:C={g¢V:9(g)<randg|,ismultiplication byascalar)uv B={9eV:(a)randg(t)=0)u A={ad(X): X€ ‘These areeasily checked tobeg-submodules of¥,included ineach other as indicated. And C/Bisatrivial g-module ofrank 1,ie.C/B =€,bytaking@ inCtothescalarAsuchthatgl,=4-1.(NotethatC/B¥Osinceonecanfind ‘anendomorphism ofthevectorspacegwhichistheidentityontandzeroon 1vector space complement tor)Weclaim also that aCcBandCoA 3) Toprove theselet¢€C,andassumetherestrictionofgto+ismultiplication bythescalar c.IX egand ¥er,then by(E.2), (X-@)() =LXe¥]~efX,Y]=0, 50X-@eB; this proves the first inclusion. IfXer, and Yes, then LY,Meet] =0,50 (X91)=-(EX,YD=(-e%YI, andX- =ad(—cX) isin4,which proves thesecond inclusion. ‘This means that themap C/A -+C/B =Cisasurjection ofg/t-modules, which must split since g/tissemisimple. Inother words, there isanelement in Csuch that ol,=id,andg-9 iscontained inA.Now let I=(Negi X-@=0}. Iiseasytocheckthattisasubalgebra ofg.Wemustverify:()1t=O,and(ii)=1+r.Forthefirst,ifXisanonzeroelementoftheintersection, then,aswesawabove,X-g=ad(—X),soad(X)=0.Hence[9,X]=0,s0€-X isanonzeroidealint,contradicting ourassumptions. For(i),letX€g.Then X-qisinA,0X-g=ad(¥)forsome¥inr.Wesawthatad(¥)=—¥-9, so(X +¥)"@ =O,ie,X+¥belongstolHenceX=(X+¥)—Visinthe sum ofland. o ThisprovestheexistenceofLevisubalgebras {ofanyLiealgebra.Wehave noneed toprove thecompanion fact that any wo Levi subalgebras are conjugate, ef,[Bour, 1,6.8] §E.2. Ado's Theorem ‘ThegoalisAdo'stheoremthateveryLiealgebraislinear,ic,isasubalgebra ofsi(V) forsome vector space ¥,which isthesame assaying ithasaFinite-dimensional faithfulrepresentation. Asintheprevioussection,thereare $6.2, Ado's Theorem soi someeasysteps,andthenacleverargumentisneededtocreateanappropriate representation.Westart,ofcourse,withtheadjointrepresentation, whichisabouttheonly representation wehave foranabstract Liealgebra g,Since thekernel oftheadjointrepresentation isthecenter¢ofg,itsufficestofindarepresentationofgwhichisfaithfulon¢.Forthenthesumofthisrepresentation andtheadjoint representation isafaithful representation ofg, ‘Theabelian Liealgebra hasafaithful representation bynilpotent matrices.Forexample,when¢=Cisonedimensional, onecantaketherepresentationAv-+(88)ingeneraladirectsumofsuch representations willsuffice, Wecanchoose asequenceofsubalgebras CMeEB) EE p= ME Bp SEDESBeaHH ‘each anideal inthenext, with n=Nil(g) thelargest nilpotent ideal ofg,and =Rad{a)thelargestsolvableideal;asin9.1wemayassumedim(q/ai-1) =| fori<q.Theplanis tostart with afaithful representation ofgo,andconstructsuccessively representations ofeachg,whicharefaithfulonc.Theconditionswwewillneedtomakethissteparethatg,=g,-1©bywithgy,asolvableidealing,andbasubalgebra ofg,.Wecanachieve thisbytaking hytobeany ‘one-dimensional vector space complementary tog.-s fori<4.Similarly to g0from rtog,useLevi's theorem towrite g=x@6forasubalgebrab. Call arepresentation pofaLiealgebra anilrepresentation ifp(X)is a nilpotent endomorphism forevery XinNil(g). Astronger version ofAdo’s theorem is: Theorem EA. Every Lie algebra has afaithful finite-dimensional nilrepresentation. ‘The crucial step is: Proposition E.5.LetgbeaLiealgebra which isadirect sumofasotoable idealaandasubalgebra bLetabeanilrepresentation ofa.Thenthereisarepresentation pof9suchthat boKer(p) <Ker(o) IFNil(g)=Nil(o)orNil(g)=9,thenpmaybetakentoheanitrepresentation. ‘Ado’stheoremfollowsreadilyfromthisproposition. Startingwithafaithful representation pyof¢=gobynilpotent matrices, one uses theproposition to construct successively nilrepresentations p,ofqy-The displayed condition assures that they areallfaithful onc.Note that ifi<p,Nil(g))=gy,whileif 1>pwehave Nillg,) =Nil(g,..) = byCorollary C26, sothehypotheses assure that allrepresentations canbetaken tobenilrepresentations. =) Supposeg=a@hisaLiealgebrawhichisadirectsumofanidealaand asubalgebra b,LetU=U(a)hetheuniversalenveloping algebraofa.Any 502 E,Ado'sandLevisTheorems ¥inadetermines alinear endomorphism LyofU,which issimply let multiplication bytheimage of¥inU.Any Xingdetermines aninner derivation ¥++[X, ¥]ofa; letDybethecorresponding derivation ofU,ef. Lemma C.27. Foreach Xingwedefine alinear mapping Tx:U+U by writing X=¥+Zwith Yin aand Zinh, and setting Ty=Ly+De, Astraightforward calculation shows that Tixs.xa=Ty9Try~Thy?Tey 6) Ifg(U)denotestheinfinite-dimensional Liealgebraofendomorphisms ofU, with theusual bracket [4,B]= AoB— BoA,thismeans that themapping 4-4 KU), X+ Ty.isahomomorphism ofLiealgebras. Suppose a:+gi(V) isafinite-dimensional representation ofa.Let4:U+End(V)bethecorresponding homomorphism ofalgebras,asin§C.3, andletIbethekernelof6.Thebasicstepis: LemmaE.7.Assumethat«issolvable.SupposeIisanidealofU=U(a)satisfying thefollowingtwoproperties:()U/lifinicedimensional;i)theimage ofevery element inNil(a) inU/tisnllpotent. Then there isan idealJ<1ofU satisfyingproperties (i)and(i),andalso(ii)foreveryderivation Dofa,thecorresponding derivation ofUmaps Jintoitsel. Granting this lemma, weprove Proposition E.5asfollows. From the representation oweconstructed anideal IinU=U(a), with U/l<End(V), 0condition (j)issatisfied; thefactthat ¢isanilrepresentation implies that condition (i)also holds. LetJbeanideal whose existence isasserted intheJemma,Becauseof(i,eachofthe endomorphisms T;ofUmaps Jintoitself,andsodetermines anendomorphism T;ofU/J.By(E.6),themappingXT; isahomomorphism ofLiealgebras from gtogl(U/J). This istherepresenta- tionprequired intheproposition. Wefirs verify that Ker(p)a<Ker(o).NotethatifXisina,thenT;is justleftmultiplication byXonU/J, 50ifp(X) vanishes, theimage ofXinU must beinJ;since JcJ,Xmaps tozero inU/l<End(V), $0a(X) =0,as required Ttremains toshow that, under either oftheadditional hypotheses, pisa nilrepresentation. Note firstthateach XinaactsonUJbyleftmultiplication, andifXisinNil(o), by(iitsimage inU/Jisnilpotent.Thusp(X)snilpotent forevery XinNil(a). Inparticular, this shows that pisanilrepresentation when Nil(g) =Nil(Intheothercase,gisnilpotent,soaisalsonilpotent,andtheprecedingshows that p(Y) isnilpotent forevery ¥ina.Weneed aslightly stronger assertion than this. LetA. End(U/J)betheassociativealgebra(withunit) generatedbyp(q),andletP<Abethetwo-sidedidealgeneratedbyp(a).The claim isthat Pisanilpotent ideal, ie,that P*=P-...-P =0forsome k.TO $€2. Ado's Theorem 03 seethis, note that there isaksuch that every product ofk elements ofp(a) is zero; thisfollows from Enge's theorem, putting theaction instrictly upper- triangular form, Toshow that P!=0,wemust show thatanyproduct of ements inp(g)which contains atTeast kmembers from p(a iszero. Butifxisinpg)andyisinp(o),wehave xy=yx+xah and(x,y]is inpa), 80terms from p(a)canbesuccessively moved totheleft‘untiltheproductisa sumofproductseachbeginningwithktermsfromp(@). Now ifgisnilpotent, foranyZinh(oring),ad(Z) isanilpotent endo-‘morphismofg,andhenceofa.BytheLeibnitzruleforderivations, itfollowsthat the corresponding derivation DzofUisnilpotent onany element, although thepower required toannihilate anelement may beunbounded However, since U/Jisfinite dimensional, itfollows readily that theinduced derivation ofU/Jisnilpotent.Inotherwords,p(Z)isnilpotentforeveryZin b,GivenXing,wileX=¥-+ZwithYeaandZeh,Choosekasinthepreceding paragraph, andchoose !sothat p(Z}'=0. Itfollows that AlX}" =(p(1) +p(Z)) vanishes, since, when thelatter isexpanded, each summand either hasp(¥) occurring atleast ktimes, orelsep(Z)' occurs Somewhere intheproduct a Tofinish, wemust prove Lemma E.7. LetQbethetwo-sided ideal inthe algebra U/!generated bytheimage ofNil(a). Since U/lisgenerated bythe image ofa,thesame argument asintheparagraph before lastshows that @*=Oforsome k.WriteQ=K/IforanidealKofU,andsetJ=K*.Clearly JcI,and weclaim thatJsatisfies theconditions ()-i)ofthelemma. Tose that Jhasfinite codimension, letx,,..., %beabass fortheimage ofain U,and choose monic polynomials pysuch that p(x) isinK;this ispossiblesinceU/Kisnitedimensional. Therefore,p(x)isinJ,sotheimagesofthexsatisfymonicequationsin U/J.inceUisgeneratedbythemonomialsxj+..."xh ifollows readily thatU/disspanned byafinite number ofthese elements. Property (i)isclear from theconstruction, forifx€Uisthe image ofan clementofNil(a),somepowerx”sinbyassumption, sox"isinIt<K*=J For(ii, ifDisaderivation ofo,since aissolvable, itfollows from Proposition C-24 that Dmaps ainto Nilfa). The corresponding derivationof Utherefore maps Uinto K,from whichitfollowsthatitmapsJ=K*toitsel.a Asbefore, theresults ofthis section also apply totealLiealgebras: ifis real, afaithful representation (complex) representation ofq@Cisauto- matically afaithful realrepresentation, andembeds gissome gl,R. APPENDIX F Invariant Theory forthe Classical Groups “Theobject sto derive justenough invariant theory forthelasical groups toverify theclaims made inthetext. Wefollow aclassical, constructive approach, ving an entity ofCapel SF.1: Thepolynomial invariants §F.2: Applications tosymplectic and orthogonal groups §F.3: ProofofCapelli’s identity §F.1. The Polynomial Invariants LetV=C%,regarded asthestandard representation ofGL,C, soofanyof thesubgroups G=SL,C, O,C, SO,C, ofSp,€ (forneven); e,,..., &denotes astandard basis forV,compatible with oneofthestandard realizations ofG. The goal istofind those polynomials F(s'.... x) ofmvariables onVwhichareinvariant byG.Forexample, ifQ:V@V—Cisthebilinearformdetermining theorthogonal orsymplectic group,thepolynomials Q(x,x")areinvariants, Inaddition, ifGisasubgroup ofSL(V), thebracket [x x. x], given bythedeterminant, al x]=det(xf), (Fa) isaninvariant ofG.The first fundamental theorem ofinvariant theory for thesegroupsassertsthatanyinvariant isapolynomial function ofthesebasic. invariants. This isthegoal ofthisappendix. ‘Wedenote by5thehomogeneous polynomial functions ofdegree donV, ic,S¢=Sym‘(V*). Foranm-tuple d=(dj,...,d,) ofnon-negative integers, letS*=$*@++@S*bethepolynomials onV®"whicharehomogeneons of $F. The Polynomial tvariants 505 degree din theithvariable, Note that Sym'(v@")* =Qs4, thesum over alldwith dy+d+°°"+dy=k,which identifies elements of 54with functions ofm-tuples inV.Wewrite F(X", ...,x")forsuch apoly-nomial,withusualabbreviations toF(x)form=1,F(x,yMorm=2,F(x,¥,2)form =3. When m=1wehave already found theinvariants: forSLC andSp,€ all syrametric powers Sateirreducible, sothere arenoinvariants unless d=0;forS0,Cthekernelofthemap$+S4-2(contracting withthe given quadratic {orm Q)is irreducible, sobyinduction oneseesthat there arenoinvariants if dis odd, whereas if'd iseven, theinvariants arescalar multiples ofthe polynomial Q(x,x}!?. (These results wllbeproved again below) Intheory onecould follow procedures outlined inthetexttodecompose thetensor productsoftheknownrepresentations S*toindouthowthetrivial representation occutsin $4,Except insmall degrees anddimensions, however, (hisiseather impractical Todescribe theG-invariant polynomials inS4,wewillcarry outan induction, firstwith respect tothetotal degree Jd, thenwith respect to theindividual multidegrees ordered antilexicographicaly: a’<dmeans that either Jj<Ed,orYd}=Ydandthelargest iforwhich djandd,differ hasd; <d, For integers iand jbetween 1and mthere isacanonical “polarization” mapDywhich takes apolynomial Fofmvariables tothepolynoral 8inOF DF)=Xs (2) This operator lowers thejthdegree by1,while itincreases theithdegree by 1,i,itmaps S*toS¢,where disthesame sequence ofmulti-indices asd,btwithdj=dy—Landdj=dy+1;id;=0setS#=0.Whenj=i,note thatbyEuler’s formula, Dyismultiplication byd,Note alsothatthese Dyare derivations: DylF Fa)=DF):Fa+FyDylFa). 3) These maps may bedescribed intrinsically interms ofthemultilinear algebra ofAppendix B,asfollows. Since only two factors areinvolved, it suffices tolook atthemap D,;when there areonly twofactors. Inthiscase themap Dx KOSS QS isdefined by oeMeBMYsoeWaSEoeMBMyoBoMe 06 F.InvariantTheoryfortheClasialGroups Equivalently, D,isthecomposite QS+HQ(S'OS)=(SK@S')QOS* +S"QS, where thesecond isdetermined bytheproduct 5¢@ S!-+S“*"ofsymmetric powers, andthefirstbythedualmapS*+5'@S*! (which takes F(x)to Yuxe @OF/A%,). Thisshows, ifthere wereanydoubt, thattheDyaremaps of GL(V}-modules, ic.,thattheyareindependent ofchoiceofcoordinates, NotethatDj,»Dymaps S*toitself. Explicitly, ford=(d,e), a OF DayDull)=Ege(Snide) or OF =Png +Danger oF wereSanger. Afirstideaisthat,ifFisaninvariantbyagroupG<GL(V),thenDy(F)willalsobeaninvariant, andtheseinvariants willbeknownbyinduction if<j, 50onecandescribe thepossible Dy©Dy(F) thatarise. Ifonealsoknew theSecondtermintheaboveexpressionforthis,onecoulddeterminee-F,whichsulces todetermine F,provided eisnotzero Ingeneral, itisnotevident how toproceed, butincasedim V=2,and 4d=(d,e), thiscanidea canbecartied through asfollows. Some oftheterms inthesecond term also occur intheexpression OF OF [y]-2F)=(2-9) (-se): Therestoccurin or ordeF=d(x23)=rWid Comparing thepreceding threeformulasgivestheidentity (4+ NeF=Day0DyalF) +Gy] QF). 2) Fromthisidentityiiseatytoindallinvariantsforoneofoursubgroups ofGL,C and forfunctions oftwo variables. WewilldoitforG=SO,C, asitillustrates theideasofthe general case—even though Gisnotsemisimple,‘andtheresultscanbeseendirectlybyidentifying GwithC*.Weassumethesimple case offunctions ofonevariable hasbeen checked: only multiples of Ol, x)?areinvariant. SupposeFeS¢@S*isaninvariantofG=SO3€,with >0,Weclaim that Fiapolynomial inthe bracket fnetion [xy] and the polynomials Q(x, y),Q(x, x),and Q(y, y).Either directly orfrom theabove — Femiy onesees that(Fis also an$0,C-invariant, andbyinduction itis2polynomial inthesebasicpolynomials, Similarlybytheanilesicographic §F.1. The Polynomial Invariants 507 induction weknow that D,,(F) isapolynomial inthethebasic invariants. It therefore sce toverify that Dy, preserves polynomials intheTour basic invariants. Bythederivation property (F.3) itisenough tocompute theeffect ofD,,onthebasicinvariants, andthisiseasy: Duley] =0, D2Ox,y)=OO, Dy,Q(x,x)=20(x,9,DaOly,y)=O. By(F-4) weconclude that(d+ I)e-F isapolynomial inthebasic invariants, wich concludes theproa Thisplanofattack, infact, extends tofindallpolynomial invariants ofall theclassical subgroups ofGL(V). What isneeded isanappropriate general- ization oftheidentity (F.4). About acentury ago Capelli found such an identity. Theclueistowrite (F.4) inthemore suggestive form Duy+LDaal|Ds, p= bo, where thedeterminant onthelefts evaluated byexpanding asusual, butbeing careful toread thecomposition ofoperators from lefttoright, since they do tetcommute ‘Thisistheformulawhichgeneralizes. IfFisafunctionofmvariablesfrom V,and dim V=m,define, following Cayley, or PF saman aca (Fs) insymbols, Qisgiven bythedeterminant aa ao ae ae ae a @ ai axe é ceé ‘TheCapelli identity istheformula: Dy+tm—1 Dia seDim DyPaxm—2oe.Dam)pangarmy. (F6) Pas Dar Dm Thisisanidentity ofoperators acting onfunctions F=F(x", ...,x") ofm variables, with m=n=dim V,and asalways thedeterminant isexpanded 08 F.InvariantTheoryfortheClassicalGroups ‘withcompositions ofoperators reading from leftoright. Note theimportant corollary:ifthe numberof variablesis greater than thedimension, m>n,then Dytm-1t Dy Pin Dy Day $mM—2-.. Daw" K=0. FD Das Dy Doe ‘This follows byregarding Fasafunction onC*which isindependent ofthe lastm~ncoordinates. Since QF) =0forsuch afunction, (F.7) follows from (Fo.‘WewillproveCapelli’identityin§F.3.Nowweuseittocomputeinvariants.LetKdenotetheoperatorontheleft-handsideoftheseCapelliidentities.The expansionofKhasamaindiagonalterm,theproduetofthediagonalentries Dy,+m ~1,which arescalars onmultihomogeneous functions. Note thatin any other product oftheexpansion, thelastnondiagonal term which occursisoneoftheDywithi<j.Sincethediagonaltermscommutewiththeothers, ‘wecangrouptheproductsthatprecedeagivenDyintooneoperator,sowecan write, forFeS4, K(F)=pF ¥PydlP wherep=(dy+m—1)-(dy+m—2)°...-(dq), andeachPyisalinearcombination ofcompositions ofvarious Dy.Capeli’s identities saythat oF=5PDIF) itm>n, 8) a pF SPD) +(x... x]-0(F) ifm=n (9) & Just asintheabove special case, ifFis aninvariant ofagroup G,each DAF) isalsoaninvariant inaS*where wewillknow allsuch invariantsby induction. IfGisasubgroup ofSL(V), and m= n,then Q(F) isalsoan invariant, asfollows from thedefinition orCapelli’ identity. Invariants forSL,C. LetF$*beaninvariantofthe group SLy€. Wemust show that Fcanbe ‘written asapolynomial inthebasic bracket polynomials. Inparticular,if m<n,wemustverifythattherearenoinvariants excepttheconstants ia 5S?=€.This isasimple consequence ofthefactthat foradense open setof ‘tuples ofvectors—namely, those which arelinearly independent—thereis ‘anautomorphism ofSL,C taking them toafixed m-tuple ofindependentvectors,say€),...yéq-S0aninvariantfunctionmusttakethesamevalueonallsuchm-tuples.Bythedensity,itmustbeconstant,Form>n,weproceedbyinductionasindicatedabove.AllDyFareknowntobeinvariants (fori <j),asis Q(F}, sothese arepolynomials inthebrackets, SF.ThePolynomial Invariants 509 ‘Tocomplete theproof, byCapelli’ identities (F.8)and (F.9, itsuffices tosee that theoperators Dyyalltake brackets toscalar multiples ofbrackets. Thisisanobviouscalculation: Daytakesabracket[xx"...x]tozeroifbdoesnotappearasoneofthesuperscripts, ortothebracketwiththevariable xreplacedbyxifxdoesoccur,thelatteriszeroifx™alsooccursand isabracket otherwise. Toavoid repeats, oneneeds only consider brackets where thesuperscripts areincreasing. This completes theproofof Proposition F.10. Polynomial invariants F(x'",..., x)ofSLCcambewritten 4spolynomials inthebrackets C8 DSi chco <iym Exercise F.11. Show that theonly polynomial invariants ofGL,€ arethe constants, Invariants forSp,C Letr=n/2,andlet@betheskew form defining thesymplectic group Sp,C, 8O(x,9)=Dien esr —SousiStandardcoordinates. Notefirstthatthe brackets are not needed: Exercise F.12*. Show thatthebracket (x!)x"... x]isequal to Fsen(oyQtx', x20. (2%, tye. (ym, ee, ‘where thesum isover allpermutations oof{1,....1} such that o(28— 1)<eifor1<i<rando(i~1)<o(i)for2<i<r. Let77betheassertion that anySp,€-invariant polynomial inmvariables {rom €*can bewritten asapolynomial inthebasic polynomials Q(x”, x). ‘The antilexicographic induction using theCapelli identities isthesame as before, andgives theimplications Trl T= Ty forallm>n. TheonlyvariationhereistoverifythattheoperatorsDaapreservepolynomials inthebasic invariants, andD,yQ(x”, x)isagain zero oranother basic invariant ‘Thesituation where m<nisalittle more complicated than that forthe special linear group, however—which ishardly surprising since there are nontrivial invariants forSp,C inthisrange. Note that 7;"implies 7,7"for in’<m, soitsufices toprove Tz". Thisisdonebyinductiononr=n/2,ke, byproving theimplication Tz} =>Ty!. Toprove this,consider therestric: tionF’ofaninvariant polynomial FonV=€*tothesubspace ’=C™?Perpendicular totheplanespannedbye,ande,.Thisrestriction isaninvariant 510 FInvariantTheoryfortheClassicalGroups ofthe group Sp,_.€. Byinduction, Fisapolynomialinthebasicinvariants. Since Q(x", x) restricts tothecorresponding invariant onV',there isa polynomial inthese Q(x", x")such thatFandthispolynomial have thesamerestriction toV".Subtracting, itsufficestoprovethatifaninvariantFrestrictstozero onV’,then Fiszero.‘Weshowfirstthattherestriction ofFtothelargersubspaceW=V'Ce, mustbezero.Fixy!,..., in’,andconsiderthefunctionofmcomplexvariables. Hlayoat)=FFeyoonHOge ‘ThefactthatFisinvariantbyautomorphisms inSp,CwhichfixV’andsend¢,t0 ae, and¢,toa”-¢, shows that Wyss Ag) =Alysetg)fOrallA#0, Since hisapolynomial, itmust beconstant, $0ty... 64)=(0, .-.40)=0, asrequire. SinceFisinvariant,itfollowsthattherestrictionofFtoanyhyperplane oftheform g°W,forany9€Sp,C iszero. Itisnothard toverify thatevery hyperplane in€"hasthisform. Soanyn—Ivectors lieinsuch anhyperplane,‘andsoFisidentically zero.Thisfinishestheproofforthesymplectic group: Proposition F.13. Polynomial invariants F(x", ....x™)of Sp,€ canbewritten 4spolynomials infunctions Qx",<), Isi<jsm Invariants forSO,C This time brackets may beneeded, aswellasthefunctions given bythe symmetric form Q,butproducts ofbrackets arenotrequired: Exercise F.14, Prove theidentity Le Ly 9 19 YI, eryen foranyvariables x... x,"05 9. Let7betheassertion thatanySO,€-invariant polynomial inmvariables ‘can bewritten asapolynomial inthebracketsandtheinvariantsQ(x",x, where wetakeQ(x,y)=SiaX:y1tobetheformdetermining theorthogonal group. The proofsoftheimplications 7")=T="form>nareexactly asinthepreceding cases, and require nofurther comment. Asbefore, it remains toprove T;-', and, byinduction on1,itsulfices toprove the implication Tz} =>Teh $F.2. Applications toSymplectic andOrthogonal Groups su Let ¥’=€™betheorthogonalcomplementtoe,.TherestrictionF”to ¥’ofan SO,C-invariant polynomial FisSO,..C-invariant, andbyinduction weknow itisapolynomial intherestrictions ofthebasic polynomials Q(x", x)andinthebracket [x"....x-}. Anapparent snag ismethere, however, since thisbracket isnottherestriction ofaninvariant on¥.By Exercise F.14, wecanwrite Fem +B(x... eM), where AandBatepolynomials intheQ'salone. Inparticular, AandBare ver, i,they areinvariants ofthe fullorthogonal group ,..€. ButFisalsoeven,sinceanyelementofO,-€istherestriction ofsomeelementinSO,C(mapping ¢,to-£¢,). Since thebracket istaken tominus itself byauto-morphisms ofdeterminant —1,wemusthaveF”=A.Thismeansthatwecan Subtract apolynomial intheinvariants Q(x, x"from F,sowecanassume F’=0, Therefore, therestriction ofFtoany hyperplane oftheform gV’, g€SO,C, iszero, Butitiseasytoverilythat(n—I}tuplesinsuchhyperplanes form anopen dense subset ofall(n~1)-tuples inC*(thecondition isthat there beanorthogonal vector ewith Q(e:e)#0).Thisproves: Proposition F.15. Polynomial invariants F(x", ....x)ofSO,Ccanbewritten 4spolynomials infunctions lx", 9) and Ext... en, With ISiS JSmI<iy <ip<o<sm Exercise F.16*. Show thatthepolynomial invariants ofO,C canbewritten aspolynomials inthefunctions Q(x", x4), 1<i<j <m. Show that odd polynomial invariants ofO,C, ie.polynomials Fwhich aretaken todet(g)- F byginO,C, can bewritten aslinear combinations ofeven invariants times brackets §F.2. Applications toSymplectic and Orthogonal Groups Weconsider thesymplectic group Sp,€_and theorthogonal group O,C together, letting Qdenote thecorresponding skew orsymmetric form. The results inthefirst section, applied tothecase d=(I,..., 1)saythat the invariants in(¥*)®* areallpolynomials inthepolynomials Q(x", x”), and bydegree considerations mmust beeven, and theyarealllinearcombinations ofproducts Q¢s$e, ey. geal, etn)... -gEaleem—tm, xm) AT) sia F.Invariant Theory forthe Classical Groups forpermutations @of{1,..-.m} such thato(2i—1)<o(2i) for |<ism? Regarding Q€V*@V*,these areobtained from theinvariant Q@:""@Q(m/2times)bypermuting thefactors.Inotherwords,onepairsoffthem‘components, [email protected] Qgives anisomorphism ofVwith ¥*,which takes vtoQ(o, —}Usingthiswecanfindallinvariants oftensorproducts(V*)®*@(¥)®,viatheisomorphism (VAP89=(VNB(VENP&(VNBV) ‘They arelinear combinationsoftheimagesoftheaboveinvariantsunderthis identification. Toseewhat they are,wejust need toseewhat happens toQ under theisomorphisms V*@V*=V*@ Vand V*@ V*xV@V: Exercise F.18,() Verify that under thecanonical isomorphism V*@V* xV*@V =Hom(¥, ¥)=End(v) Qmaps totheidentity endomorphism. (i)Letybetheimage ofQunder the canonical isomorphisin V*@V*[email protected] that . ¥=Le@en— ue forG=Sp,C,n=25 ¥=La@e forg~ 0,0. Fortheapplications inLectures 17and 19,weneed only theease I=k, ‘butwewanttoreinterprettheseinvariantsbywayofthecanonicalisomorphism (v9) =(VM (VI &Hom(v, V4) =End(V%), (F.19) In§§17,3 and19.5wedefined endomorphisms 9,€End(V®) foreachpair1 ofintegers from (1,..., 5forFthefrstpair, : HO,B02B23 Hg)=V0.29)WOW oss thecase forgeneral Iisapermutation ofthis. Weclaim that aninvariant in(V*)®%oftheform(F.17)istakenbytheisomorphism(F.19)toacomposition ‘ofoperators 3,andpermutationsin&;.Thisissimplyamatterofunraveling thedefinitions, which may besimpler tofollow pictorially than notationally. The invariant in (F.17)isdescribedbypairingtheintegersfromIto2d.These pairs areeither from thefirst d,thelastd,oroneofeach.Forexample,id=$ thepairingscouldbeasindicated: $F.2. Applications toSympleetic andOrthogonal Groups 313 forthepairs {1,3},{8,9}.(2,6), {4,7}, {5,10}.Composing before andafterwithpermutations, thiscanbechangedto o—o ‘Thecorresponding endomorphism ofV®%becomes 9,1=(1,2}.Thegeneralinvariantonegetscanbeexpressed intheform 054,09, 0°95), 0%, where oandxpermute thedfactors, andthepairs J,arethefirstppairs: f=(- 1,3). Now letAbethesubalgebra ofthering End(V®) generated. byall 9®--"@g forginthegroup G=SpxC (orO,C). Bythesimplicity ofthe ‘group, weknow that Aisasemisimple algebra ofendomorphisms. Wehavejustcomputed thattheringBofcommutators ofAistheringgenerated byallpermutations inS,andtheoperators 9).Bythegeneral theory ofsemi- simple algebras, cf.§6.2, Amust bethecommutator algebra ofB.InEnglish, anyendomorphism ofV®% which commutes with permutations and with the operators 9,must beafinite linear combination ofoperators oftheform 9@-@q for ginG.This isprecisely thefactfrom invariant theory thatwas Used inthe text. Weremark thatasimilarprocedurecanbeusedforSLC,butsinceinthis case Vand V*arenotisomorphic, todothis one must first dosome more work tocompute invariants intensor products ofcovariant and contra- variant factors. The idea issimple enough: usethecanonical isomorphism V=A™(V*) totumeach Vfactor intoseveral V*factors. Tracing through theinvariants bythisprocedure israther complicated, however, andwerefer to(Wel, 118] fordetails. Wedidnotneed thisanalysis, because itwaseasy towork thecommutator story theother way around, showing that the commutator of€[&;]isthealgebrageneratedbyallg@--@gforginSLC (orGL,©), This can, inturn, berunbackwards: Exercise F.20*. UsethefactthatthetheGL,C-invariants ofEnd(v®) aregeneratedbypermutations toshowthattheGL.,C-invariants of(V*)°4@V®% areobtained bypairing offthefactors andcontracting. There arenoGL,C- invariants in(V4) @V®"ifk#1.ForSL,C-invariants, onealso hasdeter- inant factors when k~1samultiple ofthedimension. Wealso omit anydiscussion ofthesecond fundamental theorems, which describe therelations among thegenerators ofthe rings ofinvariants (but seethediscussions attheendsofLectures17and19).Theseresultscanalsobefound in{Wel}. si F,lovariant Theory frtheClassical Groups §F.3. ProofofCapelli’s Identity Theproofsnotessentially differentfromthecasem=2,onceonehasagoodnotational scheme tokeep track ofthealgebraic manipulations which come about because thebasic operators Dydonotcommute with each other. Aconvenient waytodothisisasfollows.Forindicesiy,jy,--»ip,jpbetween1andm,define anoperator A,,;,4,,j,--- 4y,),which takes afunction Fofm variables x"), ...,xtothefunction BayeBylE)=Seca agOE Forp=1,AyisjusttheoperatorDy,butforp>|,thisisnotthecomposition oftheoperators 4,,,. Note that theorder oftheterms intheexpression Bij o>Sig,i8unimportant,‘Wecantformdeterminants ofpxpmatriceswithentriestheseAy,which actonfunctions byexpanding thedeterminant asusual, with each ofthep! products operating asabove. Forexample, forthemxmmatrix(Ay), Syl)=|SSeno):Banana +Bnei Thematrix (Ay)isaproduct ofmatrices (x)-(0/dx{"), andtaking deter- rminants gives the Lemma F21. Form=n,{Syl(F) =Ce... x)}-04F), ToproveCapelli’identity(F.6),then,wemustprovethefollowingidentity ofoperators onfunctions F(x", ...,x): Diytm=t Dy oe Dim By Bi Ow Py Diptm—2 Daal JarBr+Aamlpay Das Daa os eeeroa This isaformal identity, based onthesimple identities: Dap? Day=DypBan =BypBag itp a; Duy? Day=Byybg+Dyitp Similarly, ifp#a,forallk,then - Dey?Bays,Baa,=BapbayryosBaas (F.23) while ifthere isjust one kwith p=a4,then Dap?Bains+Bad,=BapBarns=Bad,+BarryBatyooBaa,(FA) Wherein thelastterm theAyreplaces 4, Weprove (F.22) byshowing inductively that allrxrminors ofthetwo SF, ProofofCapelli’Ldentity sis matrices of(F.22) which aretaken from thelast rcolumns areequal (as operators onfunctions Fasalways). This isobvious when r=1.Wesuppose ithasbeen proved forr=m—p,andshowitforr+.Byinduction, wemay replace thelastrcolumns ofthematrix ontheleftbythelastrcolumns ofthe‘matrixontheright.Thedifferenceofminorontheletandthecorresponding minorontherightwillthenbeamaximal minorofthematrix Dip~ Bip Bape Aim Dip S2yp Bags vsBam yp—bee+rBape Bom|" DpAmpBmp=Bm sowemustshowthatallmaximalminorsofthismatrixarezero.Supposetheminor chosen isthat using thegith rows, for|<qo <q, <“"" <4, <™m. Expanding along theleftcolumn, thisdeterminant is EoMo—E,M,+E,M, ~*~+(—1VE,M,, (F.25) Where Ey=Dap—Bapifge#Prand Ey=Dyp~Opp+Fifgg =pyand Mais thecorresponding cofactor (rxr)determinant: obs,EMMegrsetr-Searteareueseusny Serta—(F-26) Toshow that(F.25) iszero, there aretwocases. Inthefirstcase, thepth row isnotincluded intheminor, ie.,q,#pforalli.Inthis case each term EM, iszero,since E,=Dy~Sg,andalltheproducts intheexpansion of‘MareoftheformA,,y,.-ba,Withalla,p,andtheassertion follows from (F.23). Inthesecond case, thepthrowisincluded, ic.,g=pforsome k.Asinthefirstcase,(Dyp—4y,)My=0,andsinceE,=D,y—Byy+F,Wehave EM =PM. Weclaim thateachoftheothertermsE,M,,for i#k,isequalto(~1)*""'M,,from which itfollows that thealternating sum in(F.25) iszero. When M,is writtenoutasin(F.26),anditismultiplied byE,=D,,,—4,,,.anapplicationof(F.24)showsthatonegetsthesamedeterminant as(F.26),butexpandedwith theqth row moved between theq_;th andtheq,,th rows. This transposition ofrows accounts forthesign(—1)~'*', yielding E,M, = (-1/°""' My,asrequired. o Exercise F.27. Find aGL(V}linear surjection from S*@--@ S* onto V* @S11 @-- GH"! thatrealizes themap Fe+[x... x] -Q(F), Hints, Answers, and References Note:Unualyanswersorreferenesaregivenonyformoretheoreticalexer,ofthosewhichmaybereferredtoelsewhere. Lecture 1 (1.3) Thehypotheses ensure thatAVistrivial, andthebilinear map NV@A"*V+ AV=Cisaperfect pairing, ic.,itmakes each space thedual oftheother, cf.§B.3. (1.4)For(b),takethefunction atothefunction a’,wherea'(g)=a(g™').(13)YesSeeExercise6.18, (1.14) If1isaHermitian inner producton¥,letfi:V+V*betheconjugatelinear mapgiven byw+14(,-)If1isanother, thecomposite (7) «[TisLinear, anda homomorphism iHand1"areG-invatiant. Apply Schur's lemma. Lecture 2 (2.3) Forageneralformulaexpressingcompletesymmetricpolynomialsandelnen tarysymmetricpolynomials intermsofumsofpowers,seExerieA.32(. (2.4) Look attheinduced action onA*V. 27) v= U%@U™@V®,with a=b=4"+(—1p,ande= +(-0, (2.25) Answers: (i)U@®V@®U'®V'; (i)U@V" BVO Ww. (229) The regular representation willdo. (2.38)For(e)usecharactersortheisomorphism ints, Answers, and References si Home(V@W,U)=HomelW,¥*@U) (234) Schur’s lemma applies toL 235)Applytheprecedingexercise,withLpgivenbyamatrixofindeterminates. Foretal, sce(Se2,§2.2]. (2.36) Show that(,x)= 1,andcompute thesum ofthesquares ofthese representa- tions. Reference: ([Se2, 3.2) (237)Ifisthecharacterofanirceducblerepresentation, andyisthecharacterofV,let a,=(9,x"),andconsider thepower series Boe!SpecncneLylla. Sarg 5pew =Glaap Here Cruns over conjugacy classes. Since 3(C) =dim) only for C=[e}.theright- hand side isanontrivial rational function:inparticulara,cannotbezeroforallpositive (238) This isanother theorem ofBurnside. IfCisaconjugacy class inG, 9=Dyeca: V-+V isaGomap, 20multiplication byascalar ic,and2-dim ¥=Trace(@)=[Cl-24(C).Theieatealgebraicintegers,sincetheelementsPacey8Cvariesovertheconjugacyclases,generatethecenterofthegroupringZ{G,whichisafinitely generated abelian group. Now Lierwrt =164, s01GY/im V=Sic Ae-ZoICh isanalgebraic integer. Infact,thedimensionofVdivides theindex ofthecenter ofGf. (Se2,p. 53] (239) Incase thecharacter xisZ-alued, theequation .12(g)? =1]shows that|G]fsthesumof[G|non-negative integers,oneofwhich,{yeisgreaterthan1,soatleast one must be0.Ingeneral,thevaluesofxarealgebraicintegers,sincetheyare sumsofrootsofunity.Lez,»2mbethecharactersobtainedfromybytheaction ‘ofthe Galois group Gal(@/Q) (orGal(C/Q)) on7:these characters arealsochar-acterofirreduciblerepresentations ofG.Nowifz(g)#0,then[],z4a)isanonzerointeger, 80[[]ix(a)l? >1.Since thearithmetic mean isatleast thegeometric mean, Lubna)? =m.Therefore, miGl= XY,tnd?=mic, andwemust have equality forevery g€G. Inparticular, idisthedegree ofthe representation, md?=Jn(0)?=m,80.4=1 Lecture 3 825) See954 {826 IH Gisthe subgroup oforder 7,there aethee one-dimensional representa tins from G/H, and two threedimensional representations induced from H.For sencralizations, see[Se2, $8.2). sts Hints, Answers, and References (0.30) Wisembedded inthespace ofW-valued functions onGbysending we Wto thefunction which takes he1!toh-wandallother cosets tozero. Note thatif(¢x) inastofcoset representatives, themapf+.9,@f(a;givesanisomorphismfrom Homy(€G, W)10C6@en W. (0.32)For(identify theright-handsiewiththetraceofanendomorphism ofC6.For(¢,aketobethecharacteristic functionofanelementgandapply(b.(3.33)Fisthedeterminant ofleftmultiplication bytheelement«@=5.x,¢,€CGon theregularrepresentation, andF,isthedeterminant ofltmultiplication byaontheirreducibleCG-module ¥,corresponding to.Thefactorization ofFfollowsfromthe‘decomposition oftheregularrepresentation. Theirreducibility ofFfollowsfromtheicreduetbilty ofamatrix whose entries areindeterminates, using Proposition 329. Fining ginG,setthevariables x,=1andx,=Ofork+9;thecoefficient ofx,inthe determinant ofletmultiplication by1+x,¢,00F,isz(0) (834) See Exercises 38and 39. (8.38) Vcanbereplaced byV*;¥@V=Sym?V@/?¥containsatmostonecopy ofthetrivialrepresentation, IfSym?containstetrivialrepresentation, then 16h=Fzormir(@) =MEx(a?+Lav(o")) Otherwise, theright-hand sideiszero; similarly for(PV. Note thatifzyiscea,then Lxv(o? =It G41)Reference:[Se2,$13.2} (0.42) Reference: Ja-Ke, p.12. 8.43)Considertheendomorphism J@JofV@W. 0.44) For G=Z/3, therank ofRG) is2,whereas that ofR(G) is3. (G45) See(Se2,§12] fordetails Lecture 4 (4.4)Rightmultiplication byagivesamapAab—+Aba,andrightmultiplication byb ives amap back. The composites aremultiplications bynonzero scalars. Moregenerally,iFA =CGisagroupalgebra,callanelementa=Jaye,Hermitianid=&itesay-1=dyIaandbareidempotents which areHermitian, thenAab=Aba. (4.6) Abasis forKas.) =CBy' eni803. 84where Notethatey=e4044°"+tgQ,and9-0)=cif(4)=£AbasisforVcChisoy, +eWHETE0)=6;—G-1-FOFtheease3>1,use(4.10)orsee(4.43).(4.13)Notethatthehooklengthsoftheboxesinthefirstcolumnarethenumber‘sth Induct from thediagram obtained byomitting thefirst column, Hints, Answers, and References si9 (4.14) Induct asinthepreceding exercise byremoving thefirst column, considering separately thecases when theremaining diagram isone ofthe exceptions (415) Frobenius [Frot] gives these and analogous formulas for =(d— 3,3), MHI=4. (416)UsingFrobenius’formula,thecoefficient ofx'...-xbind-(xf +--+xfJeanbbenonzerocnly if=d.s0Ahastheprescribedform;thecoeticient of4-15? -..x4inA, a...) 8(— 1". (4.19) SeeExercise 4.51fora general procedure fordecomposing tensor products (420) Use Frobenius’ formulaasinExercise4.16toshowthatx(q)=(—I'z.(0), Wherep=(23~1,2y~yonus4~HandheGy.,,itheproductofeylesofengthsBove (424) If4<jusetheantiinvolution*ofAinducedbythemapgr+g"',9@Sx, notingthat2,=(ayhy)*=bya=byay,80(c4° °c)”=byob=by(a,8b)ay= °. (440)NotethatheY'sarerelatedtothe118bythesatneequationsasthesymmetricpolynomials H'stotheSchurpolynomials S's(A.9)intheappendix.Theequation(A)forthe5;intermsoftheH'sthereforeimpliesthedeterminantal formula. (443) UseFrobenius reciprocity and(442) toprove thegeneral formula. Toprove thatKatou) APY,argue byinduction ond.NotethattherestrictionofA*Vspits intoasumoftwoexteriorpowersofthestandardrepresentation, andfromanything‘butahookonecanremoveatleastthreeboxes, (44) The induced representation ofV4bytheinclusion of inSyeq 1V5©Ky Usethetransitivity ofinduction,Exercise3.16(0). (445) For(a),see[Jam, pp.79-83}. For(b),using (4.33), thecoefficient ofX*in (Of+--+ xP)Pisthe sumofthe coefcients ofX*x;" in9,summing overthose Hor which a,2m.Use thedeterminantal formula towrite z.(@) as@sum £+xa(8) adshow that the1which occur arethose obtained byremoving skew hooks. Reference: [Boe, pp.192-196} (446) SeeExercise A.11. Infact, thiscondition isequivalent tothecondition that Ky<K,.q forall p,o¢ tothecondition thatUsisisomorphic toU, W,forsomerepresentation W,ef[L-V]. (447) References: For thefirst construction see[Jam), [Ja-Ke} forthesecond, see (rey. (448)Thereareseveralwaystodothis()Usethemethodsofthislecturetoshowthatthevalve ofthecharacter ofU;ontheclass C,i[9(P*)}j,, where 9isthe involution defined inExercise A.32. Then apply Lemma A.26,(i) Show that U;@U" isisomorphic toU,-and useCorollary 4.39 (i)UseExercise 440or444. (449) Use Exercise A.32(0), (451) (@)Note that7,=Lyo,Wine andEo,=(1/2) T,e0.(z., where Syisthe‘aracerstic functionoftheconjugacyclassCi,Therefore, nak.Fn, (Ea, 00 Hints, Answers, and References {com which therequired formula follows. Forother procedutes andtablesforsmalld see[Ja-Kel, (Co}, and(Harn (6)V5Vag=Vaa0Y4®Vy =Vvwhich prove thecorresponding resusforCayADdCy...USE(a)0petmutethesubsctpts. (452) Fora),thedescribed map from AtoRissurjective bythedeterminnta formula ofExercise 440; itsanisomorphism since R,andA,arefreeofthesame rank. For(0,notethatP®corresponds tothecharacter Ya2(Cp)ta which by Exercise 221isthe clas function which iszero outside theconjugacy clas Cy,andwhorevalveonCis2).Formore ofthis corcespondence, ae(Bu), [Di2}, [Mac]. In(Kx] ating structureonthisingirelatedtorepresentation theory.In[Liu]thisHopfalgehrais utedtoderivemanyofthefatsaboutrepresentations ofG,fromscratch.In[Ze]a similarapproachisalsousedforrepresentations ofGL,(Fy)More about representations ofthe symmetric groups canalsobefound in(Fou) andUL] Lecture 5 (62) Consider theclass functions onHwhich areinvariant byconjugation byan element notin Ht. (4) Step 1.()laversesofelementsofcareconjugateoelementsofcifmiseven, andtoelements ofc”ifmisodd;x(g™')=x(0).(ii)(9,9)is2pen—eit-4were-ui)<2! yw—op. aire |u—oP+#e*-|0~ul?Y=Faeg! P (iii)IfAcorresponded top¥,thevaluesofx;andx;onthecorresponding conjugacyclasses¢’(p)andc*(p)wouldbethesamenumber, sayw,andExercise4.20impliesthat 2w=+1Since wisan algebraic integer, thisisimpossible.Therefore,2corresponds to4,and now trom Exercise420wegettheaddtionalequationu+»=(=f Step 2.(i)Information about thecharacters 7and ofX°andX°isexsly {determined from Exercise3.19,andtheatthatthecharacters oftheactorsareknown byindvetion.Inparticular,sincec')andc”(q)eachdecomposes intotwoconjugacyclasses in17,wehave _netgy=tan e+Ved Jaa -Seg reign=tte£4ee pa ~ ettene wok, where6)=(—1799,f=(—10-9, pmeyeandg!=9q"...-q,;and sinilarly fortheothervalues.(v)ThecharacterofYtakesequalvalvesoncachparofconjugate classes. (Reference: (Fro2], [Boe}). (6.5) Reference: Ja-Ke}. (59) 1f-N isanormal subgroup properly between {+1} andSL,(F, oneofthe noatevial characters must take thevale 7) kenteally on (5.11) Reference: [Stel]. Hints, Answers, and References su Lecture 6 (64) Compare (1)ofthe theorem with formulas (4.1) and(4.12) For«procedure toconstructabasisof$V,seExercise6.28 (6.10) By(4.41), thereisanisomorphism ofCS.qe-modules: CFsom®eseye024BH)®DeNagel Tensoring oftheHeftwith theright CG,,q-module V4" =VO@e V9, and toting that C(S, xE,)=CSCS, (WeVO")Beeace. (iON)=D.NinS.¥.(Thisalsosesthegeneralfact:ifA+Bisaringhomomorphism, NaletA-module,andMaright Bemodule, then M@y(B @,N) M@,N.)The le-hand side ofthe splayed equation is (7%Gc0, 4) eV" Wee, Hs) SV@SV. hich concludes theproof {6.11 (a)Thekeyobservation isthat (VO WI=DVO OW)@eye,<0y CSthesumoverallabwitha+b='d.TensoringthisontherightwiththeC(S,}moduleYoone gets WOW =DUOW)Geyeray)REM where Res, denotes therestriction toG,x6,Then useExercise 443todecompose thisrestrietion. (b)ByFrobenius reciprocity, therepresentation induced by¥,viathediagonal‘embedding ofGinS,%S,isDCieVsVy.WithACy,thissays(4@A)@yAe=DCan(Aes@Ae)Tensorthiswiththeright(A@A}-module(V @W}*™=V4@W'%Thespecialcase follow trom Exercise 451(b). (6.13) UseExercise A.32(iv),or write theleftsideasV%@Ab, anduseExercise 448, (6.14) These come from therealizations ofthe representation V,=Ac,asthe image ofthe maps Ab,—+Aagiven byright multiplication byay,andsimilarly Aay+Ab, byright multiplication byby. (615)ItisclearthatifoneallowsTtovaryoveralltableauxwithstrictlyincreasingalumsbutnoconditionsontherowsthenthecorresponding spanthefirstspaceQetoshowthatthevyforTsemistandard spantheimagethekeypointistoshow how tointerchange elements insuocessive rows. Once itis checked that the ‘ements span, theindependence canbededuced from thefactthat thenumber of semistandard tableaux isthesame asthedimension. For adirect proof ofboth spanning andindependence, see[A-B-W}—but note that their patitions areallthe conjugates ofours. Seealso Proposition 15.55, sn ints, Answers, and References (6.16)UseBxercise6.14torealizeeachS,¥whichoccursastheimagein¥%@‘ofasymmetizing map,andcheckwhetherthisimageisinvariantoranti-invariant bythemap which permutes the(wo factors (6.17) (a)Kdenting thedimelements onwhich Sixacts with thesetofpair(i,sts4.1<j<m)determinesembeddings ofthegroupsSx~~xS,(mfactors) and GyinSyq- Let =O @GE CE,B OCS,=CG,xxSCCS, C= OCS CCE. Then ¢=c'-c" isthe required element ofCS, Fora combinatorial description of plethysm see[Mac §L8]. (b)The answers are Sym'Sa.V)= Sao ®Su.2.0¥ BS..." ©Se.3.2.0¥5 BS2.9) =Sy.s,9¥ ®Sp.2.a.y¥- Reference: [Lit2,p. 278] (6.18)Theircharactersarethesame.Infact,ifxandyareeigenvalues ofanendo-‘morphismofVtheraceontheft-handsideis/(k)x497%, where(Kisthenumberofpartitions ofkintoatmost pintegers each atmost g.This number issymmetric in and g,byconjugating partitions. (6.19) Thefects about skew Schur polynomials arestraightforward generalizations of corresponding facts forregular Schur polynomials given inAppendix A;prools of {i-(io) canbefound in[Mac]. Toseethatthe twodescriptions ofVa,agreesethe hint forBxercise 44(a} Skew Schur functors arediscussed in[A-B-W], where theconstruction ofaassisgivenfromthisthecharacterformulaviifollows,ThenGx)plies x)and (x) (620)References, withproofsofsimilarstatementsinarbitrarycharacteristic (where‘theresults, however, ateweaker), are(Pet and(Jam). (6.21) References: (A-B-W] and(P-W} (6.29)Areferenceforthegeneraltheoryofsemisimplealgebrasanditsapplicationstogroup theory is(C-R, §26}. Lecture 7 (74) One waytoshow thatasymplectic transformation hasdeterminant 1,[Dil}. istoshow that thegroup SpsgC isgenerated bythose which fixahyperplane ie. transformations ofthe form vt» +1Q(0, u)forsome vectoruandscalar2.Another, of,Exercise F.12, istowrite thedeterminant asapolynomial expression interms of theform @. (72)Considertheactiononthequadric0,»)=(7.11)Forany¥,theimageofthemapx+xyx-ty" isdiscreteonlyifyincentral(7.13)PGL,€actsbyconjugation onnx»matrices, Hints, Answers, and References sw Lecture 8 (10) 0) s4EX, Y}Z)= (0X YZ} and (adX08 V2)=(adXoadY a4Youd XYZ) =O ZH) ZT (8.16)ThekernelofAdisthecenterZ(G),of.Exercise 7.11.(8.17)Usestatement(i,notingthatWisG-invariant ititisGinvariant, Gthesniversal coveringof6. (824) With A,B,C, Dnxnmatrices, 4B) cacao ="Db,4D"CB= souim)={(4 Puc='caap ='08.40~'ca =I} | out{(phen cmcn=-o} (828) Theautomorphisms ofG=G/Caretheautomorphisms of@whichpreserveC.(425)Thepointithatthecommutator oftwovectoreldsiagainavectoreld,which can bechecked inlocal coordinates. (835) Both signs aepos (438) Reference: Hol}. (842)Forhet,Morkives«coordinate neighborhood ofh.ForanotherapproachtoProposition841,withmoredetailsse(Hel,1.2). (843) Foranexample, take anysimply connected group which contains atorus of dimension greter than on,saySUS), andlake anSatta ine thera. Lecture 9 (92) IFian abaian subgroupofGandtheclaimholdsforGt,showthatitholds for6.0¢,i€Gsteazedasagroupotipotentmatrices,applyCampball-Hausdor (910) teachad(X)inilpotent,thetheoremgivesaagg=Ye>Vj9~->=0, ith, 1) Vy fom which itfllows that 9 &Ve (021) Ifhadanabelian del, emisimpliciy ofthe adjoint repetentation would ean that ther isasurjection gs ofLialgebras, Butanabelian Lialgebra hasblsofrepresentations thaarenotsemnisinple (024) Forthelasstatement, note thatthe adjoint representation isemisimple. Or seeCorallaty CL (925)Reference:(Bou,]forthis(aswellasfordetailsformanyothersateeatsinLecture 9). (0.27theadjointrepresentation isemisimple Lecture 10 have tobeanisomorphism. (10.4) Byhypothesis, theLiealgebra gofGhas anideal hwith abelian quotient;usethecorresponding exactsequenceofgroups,withthecorresponding longexact Lecture 11 (he)-£(P Reference: (B-tD, p.87), (11.19)GiventwopointsonCthereis2-dimensional vectorspaceofquadries tay heer itiesebac cite seo qudi sued yh (2ane tite rr fe (11.23) Answer: thecones over thecurve, withvertex avarying point inP?, Lecture 13 (13.3)For¥standard, Sippy¥3Tay.See§15.3fordetails (138) 0,b>0,VOMay=Tass.s@Lane1 OTana6$153. {hn Tung nesiete Vrngesesha gan sgentton Mewar tenant gene vin aed [Gre]. Hints, Answers, and References 52s Lecture 14 (1415)Thefctthat(9,4]=deoiprovedinClaim21.19 (1433) SeetheproofofProposition 1431. (1434) IFRad(a)vg, #0, then Radig) >», 2813, which isnot solvable. IF Radia) haH,and a4) #0, then 9,=(Hy94]©Radia). Use the fact that{b,Rad(qy] cRadia)toconctude thatRad(g)=Radia)vb+,Radla)og,=0.For stronger theorem, se[Va, 64.4] (1435) 16’ >6,then6">§,50'isadirectsumofandsomerootspaces9,foraT, TR, Then Tcontains some atogether with ,506>, %sl,which isnot solvable, (1436) Forsl,€, BUX, ¥)=2mTe(X0V).Forso,C,thecoefficientisn~2,andfor ‘p26, thecoeficient is(m+2) Lecture 15 (15.19)SeealsoExercise620.(1520)SeePier’formulas(69),(638).(1521)Usethedimensionformula(15.17). (1831) SeeBxerese 6.20. (1532) This isExercise 6.16 inanother notation (and restricted tothespecial linear er0up) (1533) See Exercise 6.16 (15.51)UseWeyl’unitarytrickwiththegroupUf) (1552) See Exercise 6.18, (15.54) Show byinduction onrthat r!times thedifference isanintegral linear‘combination ofgeneratorsfor':Fordetailssee[Tow2] (1557) The analogueof(15.53)isvalidfortheseproductsofminors,andthatcanbe used asinProposition 15.55 toshow that theeyforsemistandard 7”generate Ds ‘The same eyasinPropasition 15.55isahighestweightveetor.Formoreonthisconstruction, see[¥dWj;welearneditfromJ.Towber.Forotherrealizations oftherepresentations ofGL,C,ee(N-S} Lecture 16 (16.7) With ©=(ey1¢,), calulate asin$13.1; thetwovectors X53¥pX3,1¥a0 and%5.1%3,1¥a¥a0 ateproportional, andV,Xz,.Xz,,¥a¥iindependent ofthem, 526 Hints,Answers,andReferences Lecture 17 (17.18) (Note that ay:PPV+Aissurjectiveis>n.SeeExercise6.14for thesecond statement. (i)This canbedone bydirect calculation, asin [Wel,p. 155]forthehardercaseoftheorthogonal group.Or,showthat$,(V)hasahighestweight vector with weight2,andthiscannotoccurinany(V9. (17.22) This follows from thetheorem andthecorresponding result forthegeneral linear groups. OrseeExercise 6.30 Lecture 19 (193) ° Hpalat=Dorlnaet Haded=4tenn, peTandaedtOninvie) Hae Fand pet ‘TheSstaserton follows readily. Ifw=Jj0),withthe fewest numberof nonzero coeficients, and a,andagatenonzero, choose q€J\K, p¥¢JUK (possible since2k<mhthenVaq(0s)#0,Vpq(0e)=0,and80F,(w)isanonzerovectorwithfewer ‘nonzero coefficients. (19.4) The multiplicity ofLy2°bg—ba-y~~LainAVis(2iCk~~b= 21.ForFa,ofFgthemultiplicity is(2)ifispositive,bysymmetryunderreplacing any,byI,Foratheweightsare He,Ly++&La)-withe, =1,and[]=i themoltipicitesareallonesincetheseareconjugateundertheWeylgroup,similarly forT,butwith[]e;=1(19.21)Forgeneralizations, see23.2. Lecture 20 (20.17) IFfspans A*W’, and uospans Uwith Quo, te)=1,them f-(L +(17) is such agenerator. SeeExercise 20.12. (20.21) Ifxisin thecenter, take anorthogonal basis {0,),write outx=Soyo in{ermsofthebass,andlookattheequationsx:vj=+xforallj.Notethat0)= (1a C5€1,whereas04-0=(—1)"™0y-op ifJeL.Concludethato;=i isoddandthereissome j¢Icrif|/isevenandthereissomeJJ.Asimilarargument works ifxisodd, Reference (A-B-, p.7) (20.22)IX=a06,LX,0]=Hla-b-v—easy—0-a-b+0-6-4,whichis420(,Ja—a-v-b~20(4,sb+b-v-a~20{o,sb+a-0-b+20(b,da—bea) =2Q(b, 80—2010, vib=galt (2023) Reference: [Por], butnote that hisC(p, q)sourCi,p).Seealso[A-B-S}. ins, Answers, and References su (2032) IFQo—w.0~w)#0,thenRy-.(0)=Ww.Otherwise, Rysa(0)=—¥,and R.{—m) =w.For(6)compote given element ofO(Q) with anelement constructed by(a)togetonefinedonalne,andwrit,byinductiononthedimension,therestriction tothe perpendicular hyperplane aaproductofreflections. (20,33) ByBxercise 2022, X-v =(X,0].Seealso Exercise 824 (2036) Iareabass forVwithQ(0.,)) =~6,4 then@=04°..."dq: Hm=2(4), thecenter icyclicoforderfour,whilei'm=0(4)tstheKleinfourgroup. (20.37) Show thatso(Q) acts bytracless endomorphism. Forexample, thetrace of HonS*isthenumberof<(1,...,n}suchthatisevenand¢f,minusthenumber with i¢1 (20.38) For thefrst statement ofa), choose fspanning /*H” sothat, forthe chosengeneratorofPW,x(J)-e°f=fcForthesecond,whenmmiseven,x(olf=x°8°fby Exercise 20:1280Ale(shOf=tx-#-fetf)=Uef)ex60)(ES)=eS) (4)=Ble,0)TheoddcasecanbereducedtotheevencasebyimbeddingC(Q)into‘larger Clifford algebra asin Exercise 2040. (20.43)Reference:(Por) (20.48) Forexample, thetransposition ofa, andaisachievedbythematrix . pag $4 -$-t s-og $-¥-$ 4 (20.50) Reference: (Ch2, $4.3}. (2051) Reference: [Ch2, 42-45} [Jacl]. Other references include (L-M}, (Cat), (B-tD), [Hus], [P-S} Lecture 21 (21.9) If ay... rethevectors,andwehaveanontrivialrelation withnon-negative coeficients, then(0,)=Smunfaya)) <0,0.0=0.Butiesonthe same sideofthehyperplane. (21.15) The fist isruled outbyconsidering Hae, omBete teVJ6 w=Be +267+aVV/6, swith 1>(ey. eye0)Ce,W)?=1/4+3/8+3/8=1.Forthesecond,use megVmReyHeN/T, woe(Sey+deg+Bey+Dey+e@N//T5, vith64.897 (e400? 4(ety)? A41/845/12=1 (21.16) Using thecharacterization that0(H,)=8,,.onecanwritethefundamental ‘weightsimtermsofthebasis1,Thetablesin[Bowr,Ch.6]alsoexpressthemin terms ofthesimple root. sue Hints, Answers,andRs OpeWeythatythabthy 4=by+bg+bs+Silke, wontBue: (Ey: =Sik oy=Why+bythy+Labby+Le)+J3Ly, Obthetttlettsete)42t,, cgLy+Ly+hg+Ly+2/2 =Ly+Lg+Sik, watymtetbai (€) 0,=2b,opsMythathytLathy+hg+hy+SkghoyHHLy+Legthy+Latig+bg+by+Tha, Ws Ly+Lgths +Let yt She CoyLetgtLagthy+Ay p= Ly+byt Ly+hs y=bg+hey+Dy y=bythy (Fay=bythay y= 2Ly +hy+hy, 5=4OLy+LyHy+Lah only (G) y= HLy+Jka) =ay+ay, oy=S3Ly =3+Day. Hints, Answers, and References 39 (21.17) The only eases ofthe same rank that have thesame number ofroots are(By)and(C,)forallnnd(By),(C,),and(Eq)(B,)hasnrootsshorterthantheothers,(Cy) ‘ntootslonger,andin(Ey)alltherootsarethesamelength.(21.18)Forthematricessee(Bour,Ch.6]of[Hul,p.$9].Thedeterminants are: n+Lfor(A,) 2for(B,),(C,) and(E,}:4for(D,)3for(Eqand1for(G,),(F,) and(Ey). (21.23) SeeLecture 2 The proofofLemma2120sfrom(Jacl,p.124,wheredetailscanbeFound. For mote onDynkin diagrams andclassification, see(Ch3}, (Dem), (Dy-O}, {LIE}, and (Tit). Lecture 22 (225) UsethefactthatB(¥,Z)=6Te{¥'°Z) onsIy€, andtheformula (6,ef)= 3B ~diyIvo Bilenef},Z)=6-Te((3-Ey)—4-10Z)=18-Te(E,,,0Z) =IB-ef(Z-e). (22.13) Hint usetheditedeal group symmetry. Q21S) Answer: hy xAC. (22.20) For (b), note that (0 who A¥)= THe who.¥) =U0 AH) Ae) = AHL) —LWW(0). (22.21) ForaWiple I=(P<q<r} ©[ly hletemeyAe446, andsimilarly forgyFortriples Jand Ktheessential caculation (ne Exercise 225 isto verify that €72xis(—c/18)times ° twos! Ean HK=(ra.0)J ={namhmen Ege EggtBuy MEK =I=(nar ForFreudenthal's constriction, se[Fr2}, [H-S] (2224) Forst,4,€, suchaninvolution takes Eto (--1Y""""Ey.z-paa-si thefixed algebra is(X.XM=MX},whereM=(mj,withmy=Of+j4+2,and ‘otherwise my=(—1}.ThisMissymmettic if»iseven,skewifmisodd,sothefixedsubalgebra for(Apis theLiealgebra #0,q,€of(B,),andthatfor(Aza)theLie algebrasp2_€of(C.,).For(D,),thefixedalgebraiss0,4-,C,corresponding to(B,1), ‘hilefortherotationof(Da)theiedalgebraisgy.Foradescriptionofposibleautomorphisms ofsimple Liealgebras, see[Jact, (1X). (22.25)Answer:Forays€,Xi-+—X"For$034€,M25,XsPXP-,wherePsthesulomorphisiof"thatinterchangese,ande,,andpreservestheotherbasevectors. Fortheotherautomorphisms of804C,seeExercise 20.44, (22.27) References: [Her], [Jac3,p.777},[Pos],[Hul,$19.3]. (22.38) Reference [Ch2, $4.5}, [Jact, p.131], [Jact], (Lo, p.104). 530 Hints,Answers,andReferences Lecture 23 (23.3) Themap takes2=x+fytou.)withw=x/bxl,0=y(23.10)Sinceplexp(5eff)=(ens%6°".tobeintheKernelwemusthave y=2ni-nyandthenexp()a)=(—YE”. (2311) Note that thesurjectvity ofthefundamental groups isequivalent tothe connectedness ofx11) when x: G-»Gistheuniversal covering, which isequivalent{otheCartansubgroupofGcontainingthecenterof6.(23.17)Notethat'(6)=n,(H)surjectsonton,(G),andthereianexactsequence 0+my(G) +Center(G) -+Center(G) +0. (23.19) When mis odd, therepresentations aretherepresentations ofSO,C, and the products ofthose bytheone-dimensional alternating (determinant) represetation When m= 2n,therepresentations ofSO,€ with highest weights (,...y,) and (asesoes =A) areconjugate, sothat, if2%0,theycorrespondtooneirreducible representation ofO.C, whose underlying space canbeidentified with Ty,..29®Tra,..-ay- IAy=0,thenFisaniereducible representation ofOgC.Ineithercase,therepresentations correspondtopartitions2=(Ay2=2,20)See§195foranargument (2331) SeeExercises 19.6, 19.7, and 19.16. (23.36)For(by,consider(D*)?-(D-)? =(D"Dy. (2337) Reference: [B4D, VI67] (23.38) Forshysy©.TP=Fayay-tyonts-tyiFOF80240,084,TF=Taytyatant (2339) Reference: [Bour, VI §7,Bxer. 11. (2342) Compute highest weight vectors inthe(external) tensor product oftwo irreducible representations, toverify thatiisirreducible with highest weight thesum ofthe two weights. (2343) SeeExercise 20.40 and Theorems 17.5 and 19.2. (2351) Anisotropic (v~1}plane isautomatically contained inanisotropic mplane. ‘These aretwo-step flagvarieties, corresponding toomitting two nodes. (2362) For (b),use thefact that B-n'=B isopen inG.For (eit pis aweight, So'wy) =nS forxandyinB,80with xeHand w=, nsf) =fs) =flo) =Aafio. . (Other references onhomogeneous spaces include (B-G-G], [Hel], and {1}. Lecture 24 (244) (a)is proved inLemma D.25, and(b)follows, For), note thatbythedefinition‘ofpaslfthesumofthepositiveroots,»—Wp)isthesumofthosepostivefsuch that 17(f)isnegative. ints, Answers and References so (24.27) This isBrercse A62. (24.46)Thsfollowsfromformulas(A.6)and(A.65) (2451) tnthefollowing thefundamental weights arenumbered asintheanswer to Exercise 21.16 a: paBay+180,+2hay+Ha; im(hg),0=1,2,3,4%$2,1274,273,26 (Ea paLy4By+Bly+Als+4/iLg Ba, +ay +1505+Zag+15a+Ba: Aim(Fa), Lyons 27,78381, 2928, 351, 27 (EF pmly+Ly+ly+ALy+She+17/221 =Hay +4905 +6625 +964 +7505 +S2ag +2705) inh). tn, TE133,912, 8645, 365750, 27664, 1539, 56 (Gye pa Ly+My bat Aly +Slg+Oly+23g =6a; +6843 +9105 +13504 +11005 +Bag +STay +29a5; inh) =1,258 3875, 147250, 6696000, 6899079264, 146325270, 2450040, 30380, 248, (24.52) Using thedimension formula asin Exercise 249, islices tocheck which fundamental weights correspond tosmall representations, andthea which sums ofthesearetllsmallTheresultsare: (A) m2; dim Gm0?42u:dimTy,=CP thedominant weights whose representations have dimension atmost dimGare: 0,02, 01.0% 204,204, ofdimension("37%(01+0,0fdimensionn?+2n;105form=8,0,04frn=6:005,05for=7. (B,)_ 22 dimG=2n?+m;dimFy,=(24)fork<n,anddimT,=2%giving eforn=3,4,5,6 2,ofdimension10,fotn=2. (Cy) m23;dimG=20?4m;dim, =(2)~(24) ivi ono: 2a,ofdimension2n?+1; oyforn= 3 532 Hints, Answers, and References {D,) m24;dimG=2n?—n;dimF,,=(2)fork<n~2,and dimeT= dim F,,=2°,giving (4.450forn=4,§,6,7. HE) dimG= 78 01,0240 (E) dimG= 13% @,0. (E) dim G-= 248, og (Fa) dim G=32%; O06. (G) dimG=14 ones Forirveduciblerepresentations ofgeneralLiegroupswiththisproperty,sce[S-K] Other references with character formulas include (ES-K), {Kil}. (Ki2], (KI [Mur2}, and (Ra). Lecture 25 (252) Changing pbyanclement oftheWeyl group, onecan assume italodominantand2—jisasumofpostiverots.Then[2H>tnt),andcts)=(2)(oi) +=29)>0.(254)Adirectcalculation givesCX)—X-C0)=FUp(UEXT-0+L(Y,ATU. Toseethatthisiszero,write(Uj,X]=FayUjthenby(14.23),ay=(CUXI,Uj)=~((Uj, XI,Up,80[Uj.X= —F.ayUj. Theterms intheabove sums thencancel in pairs (25.6)By(14.25)Hy)=a(H)(Xa,Ye)=2X,Yo)UseExercise1428 (25.12) The symmetry gives (B02) 1g+B(=198,9)gecne=2PMapg=0 since 2(f. 2)=ma,a),sothetermscancelinpairs (25.22) Wehave Eves Wp)~pdet—n=FN"eOH(0)~od[I].a tndtheright-hand sideistbyLemma 24.3. (25.23) Wehave ZLpaper (DPW +1)—(utp—WE)OD) =EEO" L(+ 0)—1—0+0)—oh. ints, Answers and References 33 andtheinnersumiszerounlessW'(2+p)=y+NotethatfisarootofF,this happensonlyify=1byExercise252. (2524) Theminuscule weights are: (A py 24yy Oy: ow (Ce Oy (D,¥ OyDyn4sOy cr on06 (Ey ©. Reference: (Bout, VII, §7.3) (2528)Oneeasywayistousetheisomorphism s04C2s1,C xs1,€. (2530) Ny iszero bydefinition when yisnot intheclosed positive Weyl chamber ‘WandW(y+p)—pisnotin#ifW#1.Reference: (Hut). (25.40)Theweightspaceoftherestriction ofF,corresponding tojiisthedirectsumofthe weight spaces of corresponding tothose which restit tojt (2541)UsetheprecedingexecieandExercise25.23. (2543) Using theaction ofa Liealgebra onatensor product, theaction ofCon {yt isasom over terms where U,and U;actondiferent clements orthesameclement"Groupingthetermsaccordingly ladstothedisplayedformula,See(L-T,pp.19-20), Lecture 26 (26.2)IntermsofthebasisL,,Lyofh*dualto(H;,Hz},eigenvalues are+iL,and $5Ly +Ly (26.9) Reference: [Hel, $111.7]. (26.10) Constructing =go(/1) asinAppendix D,take Hsothat o(H) =H. (26.12)SeeExercise 23.6. (26.13) Reference: [Hel, §X.6.4]. (26.21) Ifaconjugate linear endomorphism g:W-+Wdidnotmap I,toitself, therewouldbeanotherfactorUofWandanizomorphism ofwithU*thehighestweight‘of({;)*cannotbelowerthan2, (2622) SeeExercise 3.43 and Exercise 2621 (26.28)References: (A-B-S], [Hus],[Por].SeealsoExercise 20.38, (26.30)Usetheidentity¥#[V]=[V@V}-20A°V} Other references onreal forms are(Gil). {BAD}. [Va 3M Hints,Answers,andReferences Appendix A {A.29) (b)UseP®=5,<H,,P™)M,. {(A.30) Someoftheseformulas alsofollowfromWeyl’scharacter formula,(A.Forpar(a)whena)24;>"ay thisit(A.19, The proof of(A) shows that forany a= (0),--4) HaMayo Hay=DBS which shows thattheK,,ateunchanged when the arereordered. Fora purely combinatorial proof see[Sta, §10}. (A.32) For(i)compare thegenerating functions E(t)=SE,t! =F](t+x0)and 1) = Ht! =YE(—1; (i)follows from (AS) and (A). For (i note that P(t)=SP=xit/lt —x40)=cHWVH(O.Exponentiate thistoget(i).Fordetails and more onthisinvolution, see[Mac] or(Sta, where itis sed toderive base ‘dentiies among symmetric polynomials. (A.39) References: [Mac], [Sta],[Fu,§4.9.4]. {AAl1) See[Mac, p.33]or(Fu,p.420]. (A48) Since 96;) =Hiand HE7) =Hi, Sep) =HU oy~Hit =toy ~Batol =Si (A.67)Answer:Ky-.--Gytimesthedeterminant ofthematrixwhoseithrowisaptSyctsnFSytorooSapient+Sacctonsn More onsymimetic polynomials can befound in(Mac), {Sta}, (L-S}, andreferencestniedin thesesourcesSomeoftheidewttes nA}arenew,althoughresultsalong these lines can befound in[Wel], (Litt), [Lit2] and [Ko-Te]; other identitiesinvolvingthedeterminants discussedin(A.canbefoundin[Me,5].Discussions‘ofSchurfunctionsandrepresentation theorycanbefoundin(Di2]and[Lit2} Appendix C (C1) Take abasis inwhich XhasJordan canonical form, andcompute vsing the corresponding basis Eyorgi). (C12) Mg=Dayand6sasimple ideat, )=[9.6]=Dla], 50iscontained insomeg. (C13) Since for5¢Derg) and Xe,ad(6(X)) =(8,nd(X)], ad isan ideal intheLicalgebraDer(ghTherefore,(ada, adlg)}~O;inparticular,if3ead(a)*andX€ then ad(6(X)) =[4,ad(X)] =0.Soad(g)' =0andad(q) =Der(q). Appendix D (D8) Toshow ad(X)isnilpotentonqe(H)forXingo(l},considerthecomplexline from HtoX:setMz) =(1 —2) +2X. Then ad{/1(2)) preserves each eigenspace (1) Wycontinuity, for2sulficiently near 0,d((2)isanonsingular transformation Hints, Answers, and References 535 ofg,{t) for240, which implies that g(H(2) iscontained ingg(), and bytheregularityofH,g(H4(2)=gf)forsmal Thismeansthattheceisanintegerksthatad(H(2)¥(Y) =Ooall¥©gq(H)andallsmall.Butad(T(2)P(¥) iapolynomialfunctionof,x04mustvanishidentically Hence, siting:=1,ad(X}?vanishesong(t},asasserted (0.24) See[Bour, Vit,83]fordetail. (D3)References: (Se3,§V.11},[Hu$122}. Appendix E Proofsofbothofthesetheoremscan befound,togetherwithmanyotherrelatedresults, in[BoutI}.Seealso[Se3],(Pos),(Va),Uact}. Appendix F (12) Check that theright-hand side ismotilinar, alternating, and takes thevalueonastandardbass.Orsee[Wel,§VL1]. (F.16) SO,C-iavariants canbewritten intheform A+TA,B, where Aandthe Aarepolynomials intheQ(x", x) and theB,arebrackets. Such istaken to Second) tem must vanish, (F20)Reference: [Wel, 1.6],oF[Br,p.866} ‘There aremany elementary references forinvariant theory, such as(D-C}, [Pr], {Spt}, and(Ho?; thelastcontains aproofofCapeli'sformula.Therearealsomany ‘modern approaches toinvariant theory, some which canbefound in[DC-P], [Sch]and[Vu]andreferencesdescribedtherein;omeofthesealsocontainsomeinvarianttheory forexceptional groups. Foramore conceptual andrepresentation-theorelic approach toCapell's identity, see[Ho3]. Weyl’ book [Wel] remains anexcellentreferenceforinvarianttheoryoftheorthogonalandsymplecticgroupstogetherwiththeelated (Br), (We2}. Bibliography [A-B]M.F. Atiyah and RBott, ALefgchets fied point formula forelitecompicresIlApplicationsAnn,Math9(1968)451-91. [ABS] MF Aliyah, R.Bolt,andAShapiro,liflordmodules,Topology’,Sopp 1968, 3-38, [A‘H-W] K!Akin, D.A.Buchsbaum, andJ.Weyman,SchutfunctorsandSchur comple, Adv, Math 4(1982), 207-278 [Ad] Adams, LecuresonLieGroups,W.A.Benjamin,In,NewYork,1968 [AhI]LV.Atlfors,ComplexAnalySecondEatin,MeGraw-Hil, NewYork1966 [AS-K] YJ. Abrainsky H,A.Jahn, and R.C.King, Frobenius symbols andthe groupsS,,GL(n),O(n),andSpin),Can.J.Math,25(1973),941-959. [And] GE: Andrews,TheTheoryofPartitions,EncylopediaofMathematicsand 113Appleations, vo. 2,Addzon-Wesley Reading, MA, 1976. [Ar] S'K-Araki Onrootsystemsandaninfinitesimal lssifcationofirreducible symmetricspaces,J.Math,OsakaCityUni,13(1960),1-34 [Aq] MCAliyah and B,O.Tall, Grovp representations, Sings, and theJ- homomorphism, Topotogy8(1969),251-297 [B-G-G) LN. Bernstein, M.Geltand,andS.1,Gelfand,Schubertcelisandcohomol- ‘fyofthespacesG/P,Russ,Math,Sur.28(1973),1-26. 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Pure Math Vol. 48,American Mathematical Society, Providence, RU1988, pp.1-13[Bove]_N’Bovrbak LieGroupsandLieAlgebras,Chapters1-3Spinge-Vetag,New York, 1989; Groupes etalgbres deLie, Chaptres 4,3 et6,Masson Pars, 1981; GroupesetalgdbresdeLie,Chaptres78Diffusion CCS: Pars, 1975; Alpebra 1,Chaptr 3,Springer-Verlag, New York, 1989 Bibliography 337 [Br] __R.Brauer,Onalgebraswhichareconnectedwiththeseunisimplecontinuous groups,Ann.Math.381937),857-872 [BAD] _T.Brécker and T.tom Dieck, RepresentationsofCompactLieGroups, Springer-Verlag, New York, 1985,[Bu]J.Burroughs, Operations inGrothendieck ringsandthesymmetricgroup,Can.J.Math,26(1974),543-550. [cal] E,Cartan, The Theory ofSpinors, Hermann, Paris, 1966, and Dover Publications, 1981 [C22] E.Cartan, Leprincipe dedualité etIathéorie des groupes simples et semi-simples, BullSc.Math.49(1925), 361-374 [cart] P.Cartier,OnH.Weyl'scharacterformula,Bull.Amer.Math.Soc.67(1961), 228-230. 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Monographs, vol. 40,American Mathematical Society, Providence, RI,1973, Index ofSymbols 9-0 =gv=p(g)(v)(groupaction, CG(groupalgebraofG),36representation), 3 Rg(G)representation ringoverK},42vomvena 1d)(numberofpartitionsofd).44 v4 42(conjugate partition to2),45 Sym*¥, 4 PaQn46 ved 4,46 Ce cy(Youngsymmetrizer), 46ma 146 VE"B®HO=a,%,-@ Hs)(powersum),48,459. aKay¥,ooKT ‘a(incriminant, 4,459 U(ivi cop.9 LID 48489 Vstandardrep)9 (Sehrpolynomia,49,454 Ualternating rep),9 A=C8,(groupringofSy),52 aycharacter of7).13 S54‘Te(race)13 Un54[o](conjugacy classofg),14 Viz54Ccam(G) (Class functions on),16 P55, 459-460 (finner product), 16 n,Q), 35,459-460 ,(oymmeticerouph 18 KE afin coads,55459-460(alternating group),18 Kye(Kostkanumber,56,456R(G)(eepresentationtingof22yn8 {externaltensorprodet),24Yyo...0%(outerproduct,$8 1,(dihedral group),30 Nig.(Litlewood-Richardson number, Chior algebra),30 58,19,82,44,427,455-486 SL,(2/3), 31 Caen OL ReagY, Res(V) (restriction of GUI) SL(F, POL CF)67 representation), 32 5,¥ (Sehurfunttor,WeylModule)76-77 IndZ¥, Ind(V) (induced representation), ys,p(9), 76-7733 SipeheFigSip82-3 saa Inder ofSymbols GLB, GL(V), AUUY) =SLR, 95 PAV(projective space offines inWY),153 2,95 (9, 153 96 [eornnp tq} 153 80,R =SO(n, SOR =SOK, 1,96 1g:PECsP*(ationalnormalcurve),154 Sp.R, 96 4,162, 198 0,8=O(0),97 a.162,198GL.C.SL,6,97 Ey(eights),163,212,239$0,697 Lyweights), 163,212, 239Spi97 ‘Aa(001lattice,165-166,198,213U,=UC,SUI,Uss=UkD, TegR,166,202,243 SU,=SUG,98 ‘Aw(eightlattice)172,200 GLH, 98 TsGrred.rop.ofsl,€withhighestweight SLM, 98-100 “aly +Bly +Ly), 176,244 p(n)=Un),98-100 Rthesetofroots.198Us. 98-100 2,(subalgebra %#1,Ccorrespondingust,98-100 toroota),200Z(G) center ofG),101 H,,lis. of», corresponding toI,X.Y PSL.R, PSL.C, POL,C, PSO.R, ins,€),200 PSO.C. PSp2.P, PSC, 102 Xa(elis. ofs,corresponding toHl,X,Y SpingP, SpingC, 102 inst,€), 2007,G(angentspace),108 ¥,(ets.ofs,corresponding toH,X,¥ rm(letmultiplication bygh,105 insf,€), 200 °¥V,(conjugation by),105 W,(reflection onb*corresponding to ‘Ad,ad(adjoint actions), 106-107 root a,200L.,JbracketinLiealgebra),107 0,(hyperplane inb*corresponding to17) ale, 109 root a),200aR,112 30(Weylgroup),201£0,R0=04R112 Vigcsteingoff),201 spa4R, 112 R"(positive root), R™(negative roots), wl 202 bats ¥(dosed Weyl chamber), 205 mR, 113 (04, (fundamental weights)aiC,sC,113 205,30620,6, 113 Fao ted. cep.with highest wt spn6, 113 Gyioy +"agen) 205 ‘ex(one-parameter subgroup), 115, (Killing fre, 206 exp(exponential map), 115, Gin} corresponding toH,inh), 2081X4¥=fog(exp(X)-exp(¥), 117 Hiy=Buy211,2392)(center ofg),124 1:Grass, -+BMY) (Picker 2%g, Mg, 25=[9,9] (commutator embedding) 227 subalgebra), 122 ,(K** power ofdeterminant), 231 Radia) (radical ofg,123 Bays 2 Se=g/Radig), 127 Vy ctgs231 X'= X,+X,Ulordan decomposition), $'V=GS*V,235-236,398 128,482 Qskew-symmetre bilinear form), 238 va SpaaC (symplectic Liegroups), 239 X47 spa.€ (symplectic Liealgebras), 239 y147 Xin iseZipUo¥(elementsin#924C), v.47 240 Index ofSymbols sas Foyited.rep.ofspawithhighestZA],375WiLay +--+ Ly), 260 (0, 3751%Gere.rep.withhighestwr ‘Chae:Rig)-+2(AJ%,375LyoooLy),260 FysooeTaitted.rp.withhighestwis0,268 (ys oh 316 Sa3h 26 AbsBB, CjDyD*D~,377-378 83,265,398 exterior power operator), 380-381 Qsymamettc bilinear form), 268 ¥'(Adams operator, 380-381804C,orthogonal Liegroups),268__b(Borelsubalgebra}, 382-3832046 (orthogonal Liealgebras), 268 (Borel subgroup), 382-383 Huays NonZu Us¥eements (parabolic subgroup), 384 in8059) 270 »(parabolic subalgebra), 384 14,296 (0), 385 5,y¥,296 Line bundle), 392ttaor,398 195396 OLY,Q)(orthogonalgroup),301 BW:8,396 C= C(Q) =CUMY, Q)Clifordalgebra), _U,U-, UW), UCWY, 396201 (1)=sen(i),400, C=COCH=COC,302 ‘(halfthesumofthepostiveroots),Ay. 50(Q),303 40 St ATI,S”=AME(hatespin C0,Y=aH)=2PAP,A,402representations), 305 ny(dimensionofweightspace)415 =AAW(opinrepresentation), 307 €(Casimiroperator),4161p, 307 P,Pye19-420 +(conjugation), 307-308 .419-420+(reversingmap),307-308 ¥,419-420«(main involution), 307-308 Plu) (Kostan’s counting function), 421 #:Spin(Q)-» SO(0), 308 Ng 428 Pin(Q), 308 HA, AIS=QU AB Spin*(p, 4)-»S0"(p. 4Spin(p,q)-> (ea form), 43080(p,gh,312 1,H{completesymmetric polynomials),(ucidean space ofroot system) 319 453 (Kling form), 320 1M,(wononsalsyrometric polynomials), ye=BAH) =28, 9),320 454 (4). (84) (Gh (Dy), 321-326 E,,Fy(elementary symmetric (E_). (Eh (Eg) (Fah (Gs) 321-326 Polynomials), 454(04,05(org),351 5,(Scliuepolynomials), 454Tas(or92)351 (bilinear formonsymmetric*telinear maps), 360 polynomials), 438 7, tilinear maps), 360 AP) 04h) 459.©(Octonians, Cayleyalgebra),362-365 2)=ig-ig!ha460 JWoedan algebra), 362-365 AGN A,461-462 H(Cartan subgroup) 369 Say. Say466-467 iced ep, with highest wt. ‘(extetiorenuttplication, 474AmAyLytonALgh370-371 TVtengoalgebras),475 Tyfy,372-373 AV(exterior algebras, 475Nenyit=90,374 ‘Sym¥(aymmetticalgebras),4752(g)(cepreseatation ring), 375 >(pairing between space and dual A= Aw 375 space), 476 6 IndexofSymboe fi(Kingformon¥),478 5(oeofsimpleroots),494 vi,430 aNil(g)=n(nilradica), 485 wmayT(corooth496 U=UG)univeral enveloping algebra), [3140] (bracken, 04436 5!=Sym),504 tun,487 AW<dfantlesicographic orden,505 eX)=explad(X), 491 4,305 Eq),491 Cayley operator, $07 Index sbetiangroupsrepresentations of),8‘Borefixedpointtheorem,384abelian Lialgebra, 121 bracket, 107-108, So4 abelian Liegroup,94 branchingformula,59,426 abelian variety, 135 Braver theorem, 36 ‘Abramsky-Jaha-King formula, 411-412 Bruhat cllanddecomposition, 395-198‘Adamsoperators,380,49 Burnside,24-25aujointform(ofaLiegroup),101adjoint represeataton, 106admisibleConcerdiagram,327 Campbell-Hausdort formula,117‘Ado'stheorem,124,500-503 Capeli’sidentity,507-508,Si4—St5algebraicgroup,95,374 Cartan,444 allernatng group (representations of) Cartan criterion forsolvability,479 %,,9 Cartan decomposition, 198,43720 Cartanmatrin,3044,29 Cartanmultiplication 29 2%,63-67 Cartan subalgebra, 198,338, 432,alternatingmap,472 478-492alternating representation,9 Cartansubgroup,369,373,381 ‘tin’theorem,36 Casimiroperator,416,429,481automorphism group ofaLiealgebra, Cauchy's identity, 457-458498 Cayleyalgebra,362-265averaging 615,21 Cayley operator, 507 center ofLialgebra, 121 character(ofrepresentation), 13,22,375, Aitneaform,40,97 440,442BorekWeilBottSchmid theorem, 392 character homomorphism, 375, 33 Character able, 14 Borel subalgebra,210,38,382 ofya Borelsubgroup,67,383,4°8 of.9 sa Index hacer becom) dimension ofLio, 93 ri 20 Sec sumo epetatons4 ore ovina sha, 4 oa ating sales, 200 wane Sodechetron id motions o 29-3 aes Somioatweia03 76at.70 iateprentiation 0,23sian Seal otsom 86coerce ea48 Dyan17characteristics (ofFrobenius), 51 Dynkin diagrams, 319-338 Chery poops howl oy 192.20chstitioni323 ,Gus Usa andsoupy12, Semyen 62,tals SeencioSahl way,79 Clebseh.237elementary subgroup, 36 {Clebsch:Gordonproblem,8,424 ementary subgroup, Clifford, 64 elementary symmetric polynomial, 77, ; “se Cordleben20,29-207,364-15ulanscommutator age sion, omersubalgebraofLiealgebra,caeepional Uealgebrasandgroups,132, ‘compact form, 432-438 0 Compl edocy¢12481-489__——$81339-35, 262-364 391-92complete symmetric polynomial, 77,453 f*S095 86?complet Liseb, 109 expe map,15-120,39-370complex Liegroup, 95 exterior algebra, 475,Comet rapeention1,44-44p —exesioralgebrnTS completors,120 exteriorpoweroferesenaton 4complexification, 430,438 ‘externaltensorproduct,24,427Conjmnteliariavtaton 436 «seta tetprodu, 2‘conjugate partition, 45,454 pecial 2-groups, conte operation64 connected Uepow, Soutacton map 18224, 20-262, Sin fandanctl hore ovarian maarear theory 3 contin 38 fed pin oral, 114393 coro 3396 fagCompete and pariah 9596,Contesgram327 Saren‘cube, rigid motions of,20 flagmanifold, 73,383-398Fou aveoe orl, £7 Fourer taro, degre epentation3 Preven93 Seton 80 489-486 Freon mip formula Served ei ‘isa Dery? Frobenius haraer fol, 4-62SnbatlformlS840406-411, Fenfecposty,337-8 pe Fede wig 28 27,25, ida grup 30,283 seam a Index S49 Gelfand, 426 King, 411,424 general linear group, 95,97, 231-237 Klimyk, 428 ‘Giambel’s formula, 404-411, 455, Kostant, 429 Grassmannian, 192,227-231, 276-278, Kostant multiplicity formula, 419-424283,286,386-388 Kostkanumbers,6-57,80,456-457,(Lagrangian and orthogonal), 386— 459 387, 390 aroup algebra, 36-39 ring, 380 level(ofaroot),330 halfspinrepresentations, 306 Levidecomposition, subalgebra, 124,Heisenberg group, 31 499-500Hermitereciprocity, 82,160,189,233lexicographic orderingofpartitions, 33Hermitian inner product, form, 6,11, 16, Liealgebra, 108, 98,99 Liegroup, 93) Hessian, 157 Liesubalgebra, 109 highest weight, 175, 203 Liesubgroup, 94 highest weight vector, 167, 175, 202 Lie's theorem, 126 homogeneous spaces, 382-398 Littlewood-Richardson number, $8,79,hhooklength(formula),50,78,411~412 82-83,424,427,455-456Hopfalgebra, 62 Littlewood-Richardson rule, $8,79, 228-227, 455-456 lower central series, 122 Icorahedron, rigid motionsof,20-30 ideal inLiealgebra, 122 immersed subgroup, 95 map between representations, 3 incidence correspondence, 193 imap between Liegroups, 93 induced representation, 32-36, 37-38, minuscule weight, 423 393 modification rules, 426 indecomposable representation, 6 ‘nodular representation, 7innermultiplicities, 415 Molien,24-25inner product, 16,23,79 module (G-module, g-module), 3,481 internal products, 476 ‘monomial symmetric polynomial, 454 invariant polynomials, So4-513 ‘morphism ofLiegroups,93 invariant subspace, 6 multilinear map, 472 irreducible representation, 4 ‘multiplicities, 7,17, 199,375 isogenous, isogeny, 101 ‘Murnaghan-Nakayama rule, 59 isotropic, 262,274,278, 304,378, 390 natural real form, 435, 437 Jacobi identity, 108,114 Newton polynomials, 460 Jacobi-Trudy identity, 455, nilradical, 485,Jordanalgebra,365 nilpotentLiealgebra,122,124-125Jordan decomposition, 128-129, 478, _nilrepresentation, S01 482-483, octonians, 362-365 Killing form, 202, 206-210, 240-241, one-parameter subgroup, 115 212,478-479 ordering ofroots, 202, 50 Index orthogonal group, 96,97, 268-269, 300, regular representation, 5,17 301,367, 374 representation, 3,95, 100, 109 orthogonal Liealgebras, 268-269 defined over afield, 41 orthonormal, 16,17, 22 ofa Liealgebra, 109 outer produc, 85,6 representations Ofe6s€y.tesfar44 ofgy,350-359,412-414 rsiring,4 ofGLC, 231-237partition,18,445,421,453, of6,146-160perfect Liealgebra, 123 ofs1,6, 161-193perfectpairing,28 ofs,€andsi€,217-231permutation representation, $ of04,273Peter-Weyl theorem, 40 ofso,€, 24-277Paffian,228 of04,277-281Piersformula,$8-59,79-81,225-227, of#4,282-286455,462 ofs0y€,294-296,Plancherel formula,38 ofsoy,312-315,planeconic,154-159 of30546,286-292,305-306,409411plethyar,8,82,151-160,185-193, (of503444,294-296,307,407-40924-231 ofspa, 244-252Plickerembedding, 227-228,389 ofsP4C,286-259Plicker equations, relations, 229,235 ofspy€, 259-266, 404-407 Poincaré-Birchof-Wit theorem, 486 representation ring positive definite, 98,99,207 ofnite group, 22Positiveroots,202,214,243,271 ofLiegrouporalgebra,375-382power sums, 48,459-460 restricted representation, 32,80, 381-382, Drimitive oot, 204, 215, 243, 71-272 425-428 projection formulas), 15,21, 23 sight action, 38-39 Projectve space, 153 root, 168, 198, 240, 270, 332-334, 489 root lattice, 166, 213,242, 273, 372-374 root space, 165,198 uadric, 189-190, 228, 274-278, root system, 320 285-286, 313,388,391, quaternions, 99,312‘quaternionic representation 41,444-489 Schutfunctor,16,222-227Sehrpolynomial49,77,223,399, 484— 462 aca,422,425, 428 Sehues Lemma, 7radicalofLiealgebra,123,483-481 semisimpleLiealgebra,123,131,209,480rank (ofLiealgebra orroot system, 321, _semistandard tableau, 56,236, 456, 461438 Serte,337 sank (ofa partition), St Severi, 392 rational normal curv, 153-160 shut, 474 real form, 430, 442 simple Lealgebra, 122, 131-132 real representation, 5,17, 444-449 simple root, 204,324 real simple Liealgebras and groups, simply reducible group, 227430-439 skewhook,99reductive Liealgebra, 131 skew Schut functor, function, 82-83, regular element, 487-488 skew symmetric bilinear form, 238 Inder ssi skew Youne diagram, 82 svipotent tices, 96Snapperconjectare, tarygrup.98solvableLiealgebra,12,125,479-480_niveralentloingalgebra,416 ‘Spechtmodule,60 486 spellnearsoup,98-97 ppangularmates,95SpoilaLicalgebra,211-212 ‘pec untry group, 98Spingroup,102,29930,307-312, Vandermonde dterainan,$9368-372 vectorfield,114 spinrepresentations, 30,81,291,295, Veronese embedding, 153-155, 189, 306,446,448 230-231,286,389spinor, 306 ‘Veronese surface, 189-193, 392 ‘spinor variety, 390 virtual character, 23,36 splitconjugacy class, 64 virtual representation, 22 Spom, 432-4345 waneweuan 18 Standard representation, 9,15, 176,244 257,273 382 standard table, 57,81 457 see 165,199 ‘Steinberg's formula, 425 weight diagram, 199string(ofroots),201,324 ‘weightlattice,172-173,200,214,242,subrepresentation, 4 273,350,372-374‘symmetric algebra,475 weightspace,165,199sammegrouppresentations of)WeksgS901 ‘Weylchamber, 205,208,215,243,256,Ss,18-20 259,272,283,292,295,351,376,495Ss,27-28 ‘Weyl character formula, 289,399-414,S31,44-62 440-444symmetric map,473, Weylgroup,201,214,243,271,340,375,‘Smetpolyoma,40-1,461-7462 ‘Wey!module, 76-84, 222-227 ‘symmetric powers (ofrepresentations), 4,Wey1's construction, 76-84, 222-227,tit,aPy-aT? powersymplectic group, 96,97,99, 238-239 Weyts integration formula, 443 symplectic Liealgebra, 239-240 ‘Weyl’ unitary trick, 128-131 wit, 365 wreathprodu, 243 tablens 48 tal, 60 panel aa Youngdiagram,45,453 tensorpowersofrepresentation, 4,472Yoursbgtensorproductoftepresentations4, 110,Younssymmestzer 424-425, 471-472 ome re, Tower, 235 Wit, 311, 32-315, 368tevpresentation, 3,9 Zak,392twistedcubecurve,155 Zein,26