Books on Group Theory

| Pooks ofGrouTHeoay i 1Pha Lucht | | | | | | i | 1 Phil's Short Bibliography ofGroup {fheory t := 1,Wybourne (1974):Classical GroupeforPhysicists. This400page,well-printed, well-written bookisaverythorough |treatment ofLie Groups. There/is much discussion ofweights, Dyakin diagrams, Cartan matrices. Thefe isaspecial section onthe 3-paraneter groups which show upalot in physios. The text concludes with three casestudies: theisotropic HO)thehydrogen ston,andshellstrusture, What isnot found inthis bookare: (1)details ofthe Poincare andrestrioted Lorentzgroups.|(2)detailsofSU(3), 2,Lyubarskii (1960): Applications bsGroupTheory inPhysics. i This 380 page Russian translation starts out with the finite groups ad all the basic group theorems. However, itdoes have 50pages specifically ontherepresentations oftheLorentz group andthe spinor and tensor algebras. Also, disoussion ofinvariant wave equations. Thebook isexceedingly dense inpriat, butgood enoughthat Weinberg references italone forthis Lorentz subject. \ |3.Ruhl (1970): the Lorentz Group andHarmonic Analysis. 300pages andmuch more specialised than the above texts, this book haslongsectionsontherepremnetations ofSL§2,C).Somehow,the 2 accentofthisbookseemstobqfunctions ofgroupspaces,Plancherelmeasures, and stuff like onewould find inaToller paper. Sort of thegeneralized Fourier/Laplacd business. Infaet, theeapoftheboek isachapter onRegge andToller poles.There isnothing inthis book about LieAlgebras orSU(3), although dotted and undotted spiners do ‘getmentioned. Not aparticularly goodplacetostart. Fr{ 4.Naimark (1964): Linear Representations ofthe Lorentz Group. Atpanslation fromtheRussian bfWaimark's 1958text. 450pages, good printing. Lote ofstuff on'spinor representations andintegral business. Concludes with discussion ofInvariat Equations.This book has too many proofs for myliking. Too specialized. t 5.Gelfand Minlos Shapiro (1963): Rebs oftheRotandLorgroups with Abpbioa. Another 1958 Russian text transjation. Lots and lots oftalk about spinors inboth the Rot and Lor| groups, and invariant equations. Thisbookwasrecommended byMN:niko. Theauthors refer toNaimark'sbook asbeing more detailed. Iagree. Looks good. i ' 6.Lioktenberg (1970): Unitary SymdndElementary Particles. fi r2 230pages,wellorganized andprinted. Openswitheomments omLie Groups and algebras, Alot ea'the Young Tableaux methods for deseribing multiplets. Dees theSU(3) Clebsch's, discusses 6-fold way, then ghapter onSUg. Closgs with discussion ofquark models.Ilike.Wa | 7.Hammermesh (1962): Group Theory. | Anice ehapter onLiegroups, Yutbeyond that itisallbasice. Thereisasmall piece about Lorents wayattheend. Donotbother with thisbook, though itisanexcellent group theory text ingeneral. 8,Lipkin(1966): LieGroupsforpedestrian. 7 Notvery organized, buthashigher groups mentioned like SU4andSU6, ‘Tyere isapiece ontheYoung dtuff, andmention ofekarm, 160pages.Obviously nomentrion ofthearentgroup. Qaaratrara) * ! | ' { 4 ' H 1 ‘ i { eBainfoutiesKelman. 7 Noieweanks “LiaoRags.&1.6." Kadgaern C162) Comtars (1266) | Sisddewtenry (149) i 1 Goudin (196%) ! . RadCaney | ® PahLeisbie ' i | )! : i - i whesksSo sunmswhoos : BeGeySuenwoaeMaDe” -Wytage ES172oneCQA2s!BY3) -ose~ ; a- | a a fo. . | -o : 7 ~ i an --a oe --—.-.- - cence ee | - -+- - | Y nw .4 e18,19,20: Iamjustgiving thereferences! forthese books here. 18.L.O.Raifeartaigh :LecturesonLocalimGroupsandtheirRepresentations Matscience Report 25,India. QA385 63!1964*Inst. ofMath. Studies, Madras-20, I. -- —~ =Well written notes: ‘Goes into some oftheparameter ‘stuff that Bargmann uses~and -+= thiswasveryhelpful. Healsodoesrootdiagrams etcwithespecial section onthe - - construction ofinvariants._ Listofother“Matecinces givenonbackcover... We"Topological Groupe”L.Pontajegin,(Moscow) Princeton UPress,1939 A385 P61939. \ This dlder book isconsidered oneofithestandard references. Contents: 7 a)abstract groups. ; b)topological spaces’ . -- ¢)topological groups | d).representations ofcompact topological groups .e)abelaincases i . f)concept ofalie group e g)structure ofcompact topological ghoups - ~h)localiyisomorphic groups(andunjversal coveringgroups)~ 4)structure ofLieGroups. 4 ~ ~~=20.Lyebl;-Group Theory andItsApplications, III--QA171162v.3PHYSThishas-anarticle by.Windternits! ontwovariable expansions andreduction ofscattering amplitudes totheultimately dymanic. "Lorentz amplitudes:". Seems _ rather complicated. | - | : \ -- |- -- - - i: i :bode:DS.Gee.onkSpaSparen. e Siqudiun Yelqasow _CoryeebeLaudAg.WelGuacee ZeBaeuns tohahoe Cue) 3.SSia's ‘ (32) | 4:Symwalic sehen @) s.decengontrnra\ ) is) (coma SS. 2 NeMumcemgadd 35. Chow SS. - a.dears, ss. to.Sedvsm.iLanilObamdanWet. MdisOksVash ChandanSota? \\Wagson @©Bk"Syrwnbicr pain”oe. ‘Hesuepng dG.fk6:G96, ‘Vrsevisabuts dHwilson prundeojealMeaWe, OnaAWaserahawibby(Helo, Daun(Gy1)ceamnnntden gr BoSHSHee\Adg.Cd)©cangnct, Jha *Riemann synpore: Cay) SUPVGH)=aahdifrediad eprnerd widaawruamrt wads Daractive AG)wdradt omO/H.SoDeakwe Coan \2wedrong. ; Warne23(930d)XEg(LA),DeDG)Na eLeo@ptpwxVER Samesaearelue iSmagaathboyef Crollonn 208(p85) GayD©DCEG/H)comVarapepeeeedoe De Doam.. FB AYdeme Quam2A1939Wasdawende DOGIH)vem! YerUl(an).Ont,NtLeeca)helena Oca.vavernonch ®U: spare Lanch~ Keeagat G = bakeomEK, exqu O/knaeedacaonobvieny - dunckia on&, ae A ‘. te4eae yet“Aesunne 8tre=Bide4, eQaawe gg week Game DR=dQVDWwDBGk). Brow AenagQuneed Lchin . OEomSemsa4:rca2)aO26) Cage!) =Als) yore3%(pa)BeckS646. guasghucack4 San FGxyy= Sasyy WryéG, .Senate mb. ©asin:cghanicadSehas(Bum63)osspacdilyWafehvao\ VO CCS~XCaR) . WweKeodorster ASadeatinaeTte KER =Hwee). Yun6-16[HortaWendel) bheeZs(sugtereds Jk9:MMe©vieWpGeenomeeoamchin do ranorgevalQueagamwiaclSetarWy e&e—=Peeng1-8)Wed] : Carruthers: "Introduction toUnitary Symmetry”:e ]‘reftQC7216291966Interscience Tracts.—§_LL _Constents: - oo oS 1.su(2). To DySUG).Cartan Weylstandard fora.VandFstarr. Weylgroupor~——— _ _ _ |. ofreflections. Regular rep=generator rep. |_3sTheIRs. Discusses direct products, Casimirs, andrelevance toparticles. 4.Properties ofthe IRs. Clebsch*s. Tensor analysis. 5.SU(3) invariant vertices and amplitues. Ratios. 6.Breaking. Both kinds ofmass formulas. Magnetic moments. 7.Crossing symmetry. Isospin crossing matrices. SU(3) crossing matrices! Appendixhasisoscalar factors. -————Added-nete-on-currentay ——- -— - — - 7 - Thisbook4snumber 27inaseries ofmonographs inphysics. Soneofthe- ~|——~ctherones_ere-of-some.interest.One_on_optical models. — — —— -- e -b -e L + one — i Si --- Sei eo > book: Unitary Symmetry andElenentary Particles D.B.LichtenbergAP(1970)QC721L486250pages db Contents:eee —~~~-—.(nupber of_pages_in_chapter)_—____1.Introduction tO" pee ~— ~~. .2.PropertiesofGroups 12 3.Symmetry, Reps, Multiplets OS i.The Symmetric Group and Identical Particles 12V— 5.Lie‘groups-and-wlgebras-—— 8B6.Multiplets 23 :--—— JvYoung—Tableaux-and_Unitary-Symmetry———— —2/,-seer____________ __ 8.Clebsh's B6V ey ——~-— -~9+TheEightfold way 2 -32 -10. The sU(6) 11 11. The Quark Model 3212,Variants ofquarkimédel ~~~-~~—‘10 1Introduction. Anpverview withsomeinteresting facts.4)saccbt,Schitz,Herglotzin1842,1697,1911 didinvariances of-class. lagr: b)Wigner showed that symtrans areeither unitary oranti. Only theunitery _—..——.ones. have conservationJawa;thus,noTawforT,orTCP. 2.Groups. Alloldhat. Some facts that areinTinkham; mention ofSygroups. 3.Grab Bag. Vector space ofQM,anilinears, antiunitaries; group representations. 01iyeheSymmetric Group.ShortdealonYoungTables.Stecopy,seenotes.~~ 5.Tealgebras. Allthis stuff isdone inWybourne. Cartan Weyl standard form. Licht-“doesbetterby‘distinguishing rootvectors fromroote; otherwise saneold-things\9 D._%,taltizlete. Thegenerators, DandFsdandf,etc.Weighttheory.Lcallsthe _ estweight. "dominant weight." Highest_weight_p_qsumtheorem.SU(3)mitiplets andtensor operators. Uses mforweight. Again, allimown stuff, donotread this.__.7sYoungTableaux andSU(n).Verynicestuff. Seecopyandnotes.Class_and triality! &.Clebsch’s. Nice, uses Youngs. Lots oftables, Wigner Eckart. See copy and notes. 9.8Fold Way. The basic multiplets ofparticles. U-spin section, theugQaxes.——2aaetclaixmissperfectUsinglet, justtothe-extent: thatUisgood——forstrongs. Does allthestandard U-spin predictions: mass splkts, Coleman ~ Glashow 1961 law, reaction-rate ratios, photopreduction-ratios,-photondecays. Photon issupposed tobe@Usinglet, hence ratios easy toget. Then GellMann~——-Okubos. Theinvariant_D andFcouplings writtenoutinfull,no_formaliam chosen. eWill copy page 178-9. 1 oe — a -— -- a ee ee | | _ book:UnitarsSymatries. 1 |Gourdin. Wiley, 1967 QC721G65 300pages “= Contents: 1.The SU(3) group. 25 _2. Physical Interpretations 7 3.Particle Classification 12 ~ — 4.Strong Interactions 35———r Eiectromguetios———- —- 26.Weaks 26 7+ Nonleptonio—weake————_-—~—-- wg8.Thesite)group. 9 94.The80(6)_model .----——-- —— 10. Ide Algebra ofCurrents 20 11.Physical Conseq's ofSU(6) 9 ~~—""""~ IeTiegroups andalgebras ToSe 713. Ide transformation groups 30 the Topologicats—- -- -3a - 15. Lie algebras ofthe semisimples 19—16.Representations ~— a -17. Tensor algebra ofthe linear group 12 appendix :isometries 000 TET havemoretimeto‘spend, this’would-bew-good place-to sportits— aaa \ | i | | | | |1464 | Tinkham : i | | | | | | } ‘ /SummaryofMajorTheorems inTinckham i} June15,1979 i ®1,Rearrangement Theorem: group rearranged byleft orright miltiply byfixed element. Jeads toHaar invariant measure for contihuous group. 2.ASubgroup anditscosets (either left orlright) partition agroup whose order ish. Thus, since order ofcosets =order ofsubgroup’= g,youmust have h/g=integer . 3.IfAisfixed, andXrunsovergroup, theXAX> generates allelements oftheclass containing A.Elements ofsame class are dalled conjugates. Identity inits own private class. CBAsses always partition thegroup. ~ - 4.GreatOrthogonality Theorem: thematrices jfortheirreducible repsareorthogonal. A 5.Dimensionality Theorem: sumof1,”=h,theorder, where1,=dimensionality ofrepresentation i. i 6.Character istrace ofmattix. Characters dftwodifferent repsareobthogonal. If two group elements are inthe same class, jthey have the same character. Thus, the rows ofacharacter table are orthogonal when weighted with class multiplicity. 7.Second-Orthogonality Theorem:theaittordhe columnsofcharactertableareorthogonal.8.Reduction: tofigureouthowarepreduced,realizethatthecharacterofthereducible repmstbesumofcharacters inthedirectaThisishowyoudoit. 9.Celebrated Theorem: Theregular rep, 1g,=h,contains eachUIR4 1,tines. @0.mumerofu'sisequaltothenumberot}classes. { 11. Abeliam groups: obviously each element ipinits own class, soh=number of |UIR's,soeachUIR,hasdimension1,| 4 i ' , ! SIT wo : Soup’ cadehogan abelian” cpoup. w=cdanhoneNoreen 8.Examples —— — TT po yg it Oo A Ty gerakeyonozoneclued Ce,ALAyAR)eacha ee ea a Nagy Say fea 2 Ba"QsicGags” sie omoust_enorabiinlles segeok. Te,SnsameCn, io FE MESas eee 3 =Ee > SSE — Uastiaqaougaloo. 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Chapter 3. definitions ofbasic coritinuous matrix groups + (like GL(n,C) eto.) matrix theorems. Chapter 4. the parameter space generators as gradient atidentity which isparameter origin. the commutator oftwo group elements structure constants and properties of the infinitesimal operators the general exponential form offinite group element. Chapter 5. definition ofalie algebra “-"~ basis’transforsiations oftheX'shomomorphisms, isomorphisms automorphisms, inner and outer; the automorphic group ideals, proper and improper (invariant subalgebras) subalgebras adjoint representations complex extensions __ simple and semisimple algebras the metric tensor orKilling form the kth derived algebra ofanalgebra asolveable lie algebra directsumsandsemidirect suns =mthe Casimir operator ofasemisinple algebra the corresponding group statements. . theorem: asemisimple lie algebra can always be written asa . direct sum ofideals, eath ideal forming asitiple glgebra. example: Weinberg's decomposition ofthe Lorentz group into two $U(2) algebras. theorem: alie algebra is semisimple iff the metric tensor isinvertible. One can always. check this directly from the structure constants. theorem: any lie algebra can bewritten asthe semidirect. sum ofa solveable lie algebra and asemisimple lie algebra. example: Poincare group, Euclidean groups. Remember that in the def ofsemidirect product, the first group (usually translations) - —-— Will beanideal ofthe complete group. -- Z Chapter 6. the Cartan Weyl basis, the Carten subalgebra roots, their properties and theorens Chapter 7. Dynkin diagrams and the Cartan matrix - enumeration ofthe basic lie algebras. Chapter 8. tle Chevalley basis 4 —i | ' I =e Chapter9.represantations, redueibility of icontragradient reps,adjoint reps,realandcomplexreps. —j theorem: asolveable group hae no finite dimensional irreducible reps except one dimensional reps. example: translations, abelian groups. There are nomatrices used todescribe translations, eg. theorem: the irreducible unitary reps ofaconnected, simple, compact group are all of finite dimension, . example? the rotation group orSU(n). theorem: the irredueible unitary representations ofanoacompact connected simple lie group are all of infinite dimension, except for the trivial rep. example: inSO(2,1) orSU(1'1), the matrices you can construct are allnon-unitary. (eg, theboost matrices) _ Chapter 10 weights and how tolabel irred reps. Chapter 11 Kronecker products Chapter 12 more onlabelling reps with weights - Chapter 13 the exceptional groups _—*» Chapter 14 dimensions ofthe‘irredreps Chapter 15 the Casimirs representations ofSO(3) and SO(2,1) compared - theorem: asemisimple lie group ofrank Lhas-L invariant Casimirs, 411 may be constructed from one formula. — -Noreover, the values ofthe Casimirs labe} completelyanyrep.lirred rep) Chapter 16 global properties oflie groups Chapter 17 the 3-parameter groups and realizations Chapter 18 applications ofsu(1,1) Chapter -19 generalized lie algebra Wigner Eckhart theorem _finglly, the case studies. - oe. 7 Nick isoLie Crow Nws com.oxprrsr. 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Welk,vinLackasPaaanainbody, hoas SSORT SOTGH Aas. taal assy] oS Ng BgSASS _— TSS LGRSS3._ST —7 __ Fh. o_ [965 \VVILENKEN | | |; | | Tayig,1977 YlenkinChapter 7 | @ verywrrortunately Toverlooked thischaptel. Itcontains michinformation Tthought was being originally presented inIBL-5527.| 50pages long, -I.have noXerox ofit. 1.Hypergeometric Furiction, 2 1, Definition 2. Relations . 3.Integrals involving 4.Jadobie (first kind only) polynomials exressed interms of.HGF. . 2TheGRoup SL(2,R). . 1,Remarks: accent here will beonldiagonal matrices 2,Parametrization. Theorem: youcshwriteg=d,pd,orsimilar, whered,are diagonal (unimédular) and piseither ay-boost ory-rotation form matrix. Close to theEuleranglesbutnotquiteduetodetajisIhaveleftout. 3,Lie algebra stated 3.Irreducible Representations..| . 1.Description: converts the QU(2) shift rep toSL(@,R).Converts tousual multiplie# rep inone real variable x. 2.Anotherrealization: Mellin'soff(x)and1)arecalledF,()andF(s). Groupactiononf(x)isthenconvetted toactiononFy(x).ThisgroupActionis thenwrittenasanintegral, therearealwaystwotermsbecause2x2sense.Kernels e ofintegrals arethefamous K,,(ya3\%ig).{ Youhave: ario R@ERO =Z .XOKO) =ZS.ayKaeafiteig) Fels Thereps arelabelled by‘X=(Iu) soweare|doing Bargmann's C,series. Thekernelsarewrittén asintegral representations hére. a 4.Infinitesimal operators. Ie, generators are written inters ofvariable x As Calculation of the Kernels. 1,Calculation oftheK'sforcasebfy-boobt (called h)ory-rotation (called u). Alsodoes for tribgulr matrices. For the ytrotation matrix hegets: he (2880~ she ahs Wath KeeQuer%h)=1POTEA22) che owct Qpimh) =ALPOMTPerrs leDeraPBNBonth“BE P28)|@s)a Bydoingac~a,c-b thingIcouldwritethigintermsofmyfunction 93)“(ewe). Bytheway,K,_=0. » 2.thé'caseoftriangularmatrices:|K'srecomputed e3.The general case. Multiply earildr results together. 44,Some hypergeometric integral trarjsforms. Nothing exciting here. 4 .Recurrence Formulas. Derives the usual recurrence formilas asinBateman by firstSatinengtheoperation ofdifferentialtion interms ofgeneratozs, then converting thegeneratosrs totheMellinspace.TheK'sareusedhere. e 6.Integral reps, and composition formula. 1.Intro: from thegroup property, the kernels must compose with noconstants atall, just Mellin-Barnes contour with adjacent "helicity" indices touching. Rest: bychoosing simple K'sinside (ie, simple g's) youcanmake lots of integral representations forthe outside Kfunction. Also, various special cases of thecomposition formula (addition theorem) aredone. . 7.Triangular andHankel: adifferent representaiton ofsamegroup. Satherepresentations, but new way todoit. Thé kennel now issome kind ofHanekel function. ‘Very short sec nothing super. Jena : Comments: Nowhere inthis chapter dowefind any reference to Azimév's work. Nor isanything said about "Legendre" .Nothing about "second kind Legendre". Whybother 4fyouareinterested intheHGFperse. Noneed toconvert toLegendre. Noonewould know that these kernels areslose toassociated Legentre functions. Vilenkin simply doesnotassociatéd theword"Legendre" withtheseocond kindfunctions: Also, there isnocomment about tieD,,*fepresentations. Ofcourse nothing ondiagonalization. Thisworkwasreferenced by‘oneofthe-Abarbanel andSaunder's papers. Theyconverted to e-functions. Iwould criticize Vilenkin for omitting tomention Legendre, and also for making theartifical separationg between QU(2) andSL(2,R). Thereal difference in these twochapters isnotthegroup butrather thebasis. Thechapter 6isinthe discrete basis andsoinvolved first kind loegendre functions; chapter 7haschosen the continuous basis and involves the second-kind legendre functions. 1— . | a, diosdoask.cone BX \ - @©@dakinonWade, eewvenins odNeue 8,NX,Ww .&. &. . akoh) PK|SEK OW=(ox|===eioecaySno -Se ih, pt=o 4ge aK aeitt ait 4itVy = (te hhe\ 2 AERo = | ees(Soaat)©| mt= TakFRWeaKo=errdom. |SeaMame#we)= +=. [u@ =os Sows gate =N=-ief\= -3K a=ik, ae ER =AK ' a= ak |.ariy= | -[Os=ide eh ®Nott,§rsod | .4k} £& -.Tcl =Tle ‘\=< fly pte te ee-% ae =4 aoe - Que: JA=x - Ac= ik. - Ay=ida, @ We=-A sche =—Gek) =c(ih).= ik=Ki. e "ye WL=be[ee-351 Keenile =i€°legs 4Bel e ee ee-ert te os“|YeeielargstFe] |Wy=-e+b3pe ° 8@=@*Zane aa w=OTZaz “Oe @DSandor groduch: -° | . -2a- - 2 pewiadee) ¢= - [xGd.=WaryotGeetehfe3@) ® Gbesa- =— 7 Gd= SLE &YA 9Sde =)4M. =Naeen,er wo . -aysae(pena)ioeon | 7kee SFakesree. -aa own| Sane Qume- Aw(8)=wee=pesGED ;rampese or(Fy97 -Le Jenre x/ / ; Tere-—=(Rees) Oe [tnG)=ayake (grea (Baty aw t=}-6R - ah _—= an vy , _ keCasy Grongsede ee EEMe OE (Brat\.s FCs) k-CEds7 © _Cpeee\ =(0-8 \&I) ; San aydeny Sey \ast LOREOR yeBETZ IOL) -woe Terme]- 1Ete Jancre __8ds4s) ‘(a-s\ft) @-s) YkA=tm) , - avn! jew! eYn)_eed gyn" us - OP Letm-d]t Bearegyiedame€ ASsee ay =EN) Sark, :bo<-liw e-b-\=f-n-€a= -fre-m arb =Q-n~esl a ee lone esKingk eye lom =omens - Se.udange weDNC-Dew)Maw-ensuew(e-Rem)- miVQamaen) np(~awLeLew) ; KwECReer nyeke Lanta, Glmw) -3- ~QedsooneQekS: -- dom! - Kipp Goan@|cya PARah Cy FNG Manenar) siantan). \ — ;. TQ-mees) "ion(iwLe“Aewil) ECRem,ehiemychew! \ay) _. Spm Sant Qf!Ema) =BY GP Gay TdawS PG) guacr(bhew orl J. - TO-m'ewt ‘ e@1vecoBeco!y\epeod 5\eN)) BDMeowaNomar Ce)deSpamDae: teVE *ee . X/ rc_T vatsa1’own 6FG dos\Paawy \&\)>an nee >, oe Pai aedSeed @3)Cane) =OR Mads [Ee eae GXRae (araw) (aeiery’ |Po Cale *ees (ree) ‘fsCaistet) CS AsIseeit,thieresultmustbetrueforallmel"andfor(mtn')oneitherlattice. Afterall,weobtained thisresultfromVilenkins general representation, _ @ ©OMedWHvowa.gormmargaiir: Ondua)bwaWe: .- AC) -tS).x=MEC b=het_© —Van ehayell= ke=2 . . re ec CeKan =Gs)" © Beas RRL) JPGGOBA Cr'-~!(agy™ oP e 7@sao" at Kisi-%) \ ‘ 7(seyNea MOY dewiryoa! 8 ~e = vag’(SAT EmYates fa or) } C@eacn shoe(®)=yot™Wace.ee)Pee(a2)@an Riko CFM TK. a ~\- | 7@KodeadeAgayeZiwask S=~$t meOu Ose CAH =Hoa.andalod ; (geez)=Coxe) =AC\-s)=7(~Vals)7 -(Bat)=@-#s=Bod - - Sevakmamcpa=as(EPS. Gye sity @PGaye” | =@"ay Cah figsSEGasyatycat”- NX fF OO © Golone +cand Oran _ - . B Waa,geking KC,pMOuLemrstude! x Q TWal-w (dr) + ala! BE9GBtemdadeOesailse! - ‘®Vascleconsisted) 4NYCy,(3)galsok. - ac em)BSCoar\2BomCone) ( n :—_1_. PoaCAC)=ao(ot)=Vs)Ry(a)PQA -») Fae5)ml OeRaa T" Q A tentmo rage Rade)=GLBa=.Wises)Ge(ake.—Pele Oey en : a )weArt (Us)Ry eee 1PQA) Yh) R= ay Ure dP Gata1S ae(a) (=)A—HGNRQsbom)=a(tae5) Ugh.*PComemayel) Ope ; neOda eT=) AKT Pa)" : Ow3seesve . -¥. -u 7ee = UQa\- u am*ive)() Sorekpon Ss rlvit. . J :roarorpyl BAe {aUeCSEaa5\* —\ Wan & e - ze7rat~v)Gul . Siwae u > &@)=gplBe" |. —s—©BakSmaweaanetiaonet sl ; -+e BN eves Koh e Att - x) Ypaey\ =GypA)(Rl=ion)_ . -.S Q ve Q Be “Rae =ARE) asce) yen| ~@WA,dakweefr%,. ;vit am - d .; Taya xraeKno(2)m0. — RbfaARweVane 7 - my AE ofWe =Fro(2) "~ “But this cannot betrue because look atVilenkins symmitry rule top ofpage 124. This shows clearly that his P's are not the same jasAzimovs. Therefore, -hemust have . — faked usout byusing different Jacobis. Hig Pfunctions must bemore like my.d a6functions. Sothisthrowslotsoftingslff. Comment:IsimplydonottrustVilenkin'seretions becauseherefusestostandardize things using hypergeonetric functions. Ihate thet, nowaytotellwhere youare.So Ishdiildreallyonlyusehimforkdeas,notrresults. |- - ai-a . H Comment_onaCommonErrorinGroup/Theory: | >: 1Usually, thestandard grouprepforyulareads: 7-7- a C3ES0 —_:One,atetheerronefous impression thetgo"willbethefiretgroupelenent . _.. te"touch" x: le . re . — Bo 2TOOT) HO=T|e ax. -.— :ToeSaige).-- These. equationsarewrong,InfactsBristhefirst,groupelementtotouch— ~%2.Theaboveequations aresimply almisunderstanding ofthenotation. The__|. bra-ket notation makesthisveryclepre __ ,STAD =Cabal»ia.Bearnanning,TDSC)=$Gx). ©.SerSENT SO),ielTat(9) SH dw = TOFD =GND BD=<xTd =LakRP=<aigixl a<xLaad'x RQ=<jgax\ >.-oe _we ALT GAD AGTQgod. =Tego4 &). . Baseat: xlTEATEIN =f<gtx(TGs i | RKmake Ghdock xJost=hax\P> =Ce@gax|> =SxlTq|: .. | - os” | t e Gonment_on Normalizers_ chVilemewy, 1.SofarIhavebeenthinkinginsexksofhavingcompletefreedominchoosing™s"normalizer Aqfor arepfesentation. However, I’now think certain choices are more natural than others. For example, look at page 307 ofVilenkin where hedeal§ with SU(1,1) matrix elements. There, equations (1)and (2)afe updengably true: these equations give you the representations irred{cibles. Now, ifyou-choose basis functions inthis space which are normalized and otthogonal, ie, swe”- $m)=.fon€ee - s+e well, this choiceisorthonormdl with.respect_to.the simplescalar product given in(4) page 304. whan you dothis, you get avery explicit matrix element without any "normaljzer freedom". See (4). 2.Howthencanweunderstand the404ofchanging normalizers? “TNS ime soneeene -fuls\= Ane A da. 3 se vn= ow : \YorCaso =1.gou (Van nuded. Bubqo e coudatinakoote/ 1en) |- “\ A Selher=ZTGuest ~*~Toagesr *2Qeioe OK,thenallyouhavetodoisretiefine thescalar product tonormalize your basis sates. In-fact, now look -at-Viienkin~page-118 to-see that this isexactly what hedoes for SU(2) jpatrix elements. 1 . 3.Thus, here iswhat ihave learned First, for SU(2), Vilenkin used the standard normalizer thateverybody uses.Theonlyotherthingyouhaveto “knowisthathechooses exp(+i0 U4)forbiscentrel groupelenent thus 7obtaining therotation matrixfunctions whichhecallsPJpn(z). Second, for SU(1,1) inthp’continuous series Vilenkin chooses theunitnormalizer A,=1.Then,hsshownonpage309,hechooses for hiscentral element exp(-itK,) «|Theresulting matrix elements arethen; 1" called Bm(cht). thie _ 4.KeepinmindthatAndrewsand1useJzastheircentralgenerator... 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Que: A=xEC) i 7 ©Garnannante :WaSomaImeoo kum Yo UG) UA. . ®.Qa araate bA, woMilerPX x22 ©Berea ArdegkWelWoauh Hy.Qaoee 8 Wee>Oe Ha= P-xhe==Wyege : : QnxPlerenmasigunbumcten ofdio.Hayste,42Cu).Cx) ond (13). . - eee —~.- .-@ e e | » or Sashiow Ir,2.3 dus shady) — ‘lets review this magic show startinglwith section 2.1. First, weshow that you e -cansetup.agrouprepresentation T(g)which{is anoperator acting onafunction of 2.complex variables (forSL(2,C) ).Thisoperator takesyourfunction oftwovariables andgenerates anewfunction byreplacing thsevariables withnewcomplex variablesobteinedsimplybymultiplying theoldvariatesbackwardsbya2x2matrixrepforg. Oddly enough, westart with a2-dimensional datrix representation ofgroup Gand from thatweconstruct amoregeneralrepresentation asanoperator onafunction space. Since ‘this fun¢tioni space has invariant subsfjaces under T(g) consisting ofhomogeneous polynomials ofdegree 2L,weconclude that T(g) isahighly reducible representation. Infact, we-are easily able todeduce the effect ofT(g) onafunction (2) whichisrestricted totheHilbert subspace Hyofdegree 2Lpolynomials. Thisisshown inequation (8). Inthis sectdon 2.3, weshow that thdrepresentation Ty,does notitself have invariant subspaces andistherefore irreducit le.Later,wewillshowhowequation (8)canbeused togetthematrix elements ofthdirreducible reps insome basis. SoTE goS du A - < . - P Qkx ~ite ith ©oCoupde. TExB=(e:(cho \ ore‘sikh teMe nit 0=©. (ex) =e xX4 . -Cea, Bed)=CTC) AA, FO)G9) .\ | QuaWeelacamer. Than4,0YYQaswsMak.sucka8.9. .- FakJRO , @Mua,QShinggh,WALonnaude |GOO =O,iaOre e@Byterry EK) =Gt)=2aOnegshe+ Be de 7GESOF)=Cake)!Gale)!| .Bok. Q MR) = Gren (VEG WES) =1) ) (Qrey! (Reh, y \it.g 6 Q2)S:2.A\Sxe's Q2 _ ©WHwsen Or en ee renee Pace onexoaSUC)wedsathwillsDHL, -=vaaeod dracortespace Ox. Saw. on. coe]Soe shanewernctine _. 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Hermann { | —‘ AofHermann Chapter 3. “ ASUREDofHermannShaper35arTnteqets,ondPQ” [email protected]? thepurpose ofthischapter istolearn about therelationship 7 between the first and second kind functions} AsHis starting ‘point; Hreminds us ofcertain ‘simple Tormulus whith rélute the|P, and-Q; functions. Heshall generalize these Neuman formlae. ‘The trick for doing}so isoutlined: you want your e-functions tobematrix elements. ofanoperator -D(g) insomeappropriate. space. _Forexample: Ter Gt ee oo -- ExCay =<oelBG)KP . “ me) When label jtakes special values (integers|.eg), the vectors shown are supposed to space acertain space HJ.This isthe_usuallspace connected withDJ,ie, - + -" s - -.t| Vay =ZoBROW _ (2) Now suppose ithappens that these basis funttions are-not-really the usual things, and that(j/k')isnotJustaKroenecker delta,|but rather, suppose thisquantity. is |divergent, or-otherwise poorly defined. Ie} wearenotuseing @real scaler product here,justabidinear form.Clearly, then,¥edonothaveanorthonormal basisin ourspaceH'ofdimension2j+1.Thereforg,youcannotsimplysubstituteequation e@(2)into@1)andconclude thattheEfunctidns arethesameastheDfunctions. This wouldbetheconclusion ifwedidhaveanolthonromal basis,sowehadbetternothavethat. . - . Although youtannot put(2)into (1)asshown, youcandowhat, follows: Ven enicg i Vent) AN AERCPQsGIOGd fAVE) DQ .. + . ~-=" \,=KDS)KDpw @) Pause before finishing. Thefirst step follows because D(g) issupposed tobe agroup representation -byoperators in-our space. The second step involves some kind of"adjoint" operator with respect tothis crazy scalr-like product. Nowfi ° from (2), taking the adjoint weget: wrongl Instead, thenext step isthis: youcaneasilyprovethattheadjointobjectisbasearepresentation, theonethatHcalls D'J(g). Itisthis representation-which issupposed todo(2)above. If- it”does,thenyouceninsert(2)(withaprihieon"D)into(3)toget: - ar ae ~omELDag"Cay)GeIB@VTED e« .w . \oe 4 odes _ > : S 7a 3S T 4 ~, So,whatHisshowing us.is howthat“group-like property" might coineabout. we repeatthatthereissomething rathertrickyaboutthescalarproductlikeform. e 2.General Remarks’ about Cauchy Kernels forComplex LieGroups. Here weobtain amethod of‘sort of“extenting" afunction ohGtoanew function onG,(minus G,however), where G,isthecomplexification ofG.The newfunction P(g.) isgenerated byintegration oforoginal f(g) over G,weighted by@special so-called "Cauchy'kernel". . By,requiring that-this kernel havethespecial formG(g.+g)=.C(gg,), thenewfunction F(g,) bears aspecial relation tof(g), namely: iff(g) maps intoF(g,); then£(g,g) mapsintoF(g.g,)- Somehow, left~translation "shines through" this"transformation". Righttranslation doesnot,however. (Here, 8 ‘mstofGoiirse beifG).” 7 -~>Next; forreason-unknown, H-points ‘out-that-you can;“if-youTike;“alsoinsist that the your Cauchy kernel-be "invariant undér AdK'* whereXis‘somesubgroup ofG. ‘Thenotation AdK*means the set’ofinner automorphisms ofGwhich aregenerated bytheelements ofK.Ie,gmapsinto-kgk isanautomorphism. “invariance under Ad'K"isafancywayofsaying thatC(g,)=0(kgk™). e Ifthis condition isalso met, then not’ only will your cuachy transformation beleft-translation "invariant" forallG,itwillalsoberight-translation’~ * invariant TorXK.Ofcourse'thisimplies thatforK,tlemapés‘rilght~ andleft- . translation invariant. The significance ofall this is: inthis case, iff(g) isaspherical function, thensoisF(ge). Wehaveinmind,ofcourse, thatf(g) islike-P;(z) andF(g¢)- 4slikeQ;(z¢). For the reconrd, the phrase "spherical function" means two things 1)thefunction iseigenfunction.of differential casimir | . 2)thefunction isleft-and-right, translation invariant. Clearly, PyandQ;are both spherical. oa 3.Hua's Constfuction ofCauchy Kernels. . This isa-strange digression which israther vague. Apparently, Hua has amethod_ which gives.you aformula for the Cauchy kernel for G=GL(n,R). H claims, without explaining why,thatthisnicegenerel method simply doesnot e apply tothe Lorentz group cases. Hua was 1963, bythe way. ‘ _ Oo oa . .Wignerd-functions ofthesecondkind.{ ; -Cee _ ~ - ry Iwillsimplystatethegistofthissectionwhichisveryclear.Suppose “""youactuallyHavesomeDandEfictions witcharérelatedasshownin(4.6)and -Au.7) (the-E and-D functions-are so-related!)./ Then, youcaninvent operators - By(g)-andD,(g)inobviousway.Ttfollowsallinreverse,—that -theseoperators .arerelatedby.theCuchykerneltranaformpttion shawnin(4.3),andtheactualkernel Atself 4sgivenin(iy1). However, [email protected] efunctions, ,Thepointisnotthatthere?somefancycompactformforthisCauchykernel.Thepoint,isthatthereexistssuchakernel,andtheDandE operatorsares@Telateds|esau.wt[fem Ls 5sGroup theoretic meaning-of the-integral tepresentations. Hereab.last we.get.specific. Rirst!# quotesthestandard integer] reps_ fortheassociated PandQfunctions andnotes that. they look similar, Then -hesetsup.astandard(tome,bynow),realibation oftheSL(2,R)liealgebra |as6ifferential operators inareal“variablé whichhecellsTHETA, butwhich,is,@verymehLiketheqvariableofMuiunda,foiethatyoucantalkaboutthese °™ generators andheir eigerfunctions without! using anyscalar product. Next, , hesetsupwhathehopesaregoingtobs“tde-functions bychédsing @particuler scalar products suggested bytheintegral répresentations He.sayst..Sag 2S50 Se. aw estVeeCRY=Sac_s™ expfth®) . (x) -Cet SM by Themost important thing tSedlide “at‘the atart ib’that these arenotthestates /m) we‘are uBbd''to. |These aieactually ‘states Which diagtrialize oneoftHehyperbolic gene¥ators.’ But, heistaking this generator! to..be anti-hermitian (formally, and alsowrttohisscalarproduct). Thatiswhyhegetsm,whereIwouldgetip.Heneversayshismisreal,butIsuspectitsupposedtobereal.Therefore:Lali SSS GROOM) |=nadia bESage!-sAfterthis,Hcomesupwithaveryscsiforexpoentiating thedifferential e gnérator. Thiswasaproblem Icouldneverseemtosolve. Iknewthatfirstorder-diffgensalwayscorresponded t6multiplierToe,‘butcouldneverdoitsoexplicitly!!! - | -- 80, having shown how tofind the multiplier rep, _Hproceeds toshow that theintegralinquestiondoesinfactleadfousfunctionQ,™initsusualintegral =)rep.Sohuuray, hehasexplicitly written theQfunctions asmatrix elements of certain operator insomespace. Backinsebtion 4wehaddone thisformally, but weneverstatedanythingexplicitfortheoberators DandE,andnothingexplicitforthescalar product.” Here, things areméde explicit. . Next,wechangevariablestowalorasHsays,uo.Herewrites thegenerators inthe-new variable -andnotésthatthese-are theones-having todo- withthelinearfractional transformations itheu-plane.ForSL(2,R),these map the real line into the real line, and the circle into the unit circle. Thus, either ofthese sets oftheu-plane isagogd cancidate forascalar product (because, functions-on-the-real-line aremapped into functions-on-the-real-line, etc. We have already used thereal axis togetthe rep. Itturns out, asonemight expect,thattheunitcirclegivestheintegralmeleethePfunctions,andHshowsit.Thissection isthenclosed outby#computing theexplicit "adjoint"rep andshowingtheD=DEadditiontheorem.| - e6,Thetechnique.ingeneralterms. _-| _ Iamnotquiteuptothis,butwillleyanyway.YouhaveagroupGand asubgroup L,andsomehow thisgives aspacelG/L whose points aredencted simply byp.Thismustbethattransitive spaceMpeVileniinmentions,thecosetspace. Recall thet allhomogeneous spaces must be Hpmomorphic toG/LYorsGifie subggroup Le So,nowmake upsome functions onthis M-space, f(p). Then, make amultiplierrepresentationaSshownincoil(6.1). Then, take everything complex, so thhtMgoes tospace M,=Go/L,- Then inthisspaceM,,takesomeothermanifold, tallStN,acertain lociofpoints PeUsethismanifoldtodefineascalarprodust|aeshowninp113. IannotsureIamgetting thepoint! Itsoensthatyougenerate "second kind" functions bysomehow changing thedomain ofyour scalar product. 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LM ke, e& Dada daneck ke.Qe ay, Ohetden teHLKOd amdwade bseoggana, SeiCy)=3G) RD=orunbegul aedkasun. : OK,wieshaglaceobedharmafeoh Weeats owSeyEC)awoe eYeWaFn Bodenack. Qhogher a: powAsathong. Bowe DTM Ay=ee)a"oeVodue4Sumcrvensh awQuseknde Ce=Stad* e UjonanYpmrelrsoh (aa)‘aa)=etayy a —8-. wkABA Suge Aten obec.Qh O |Gav= aro _.. Be . ; a. AaGes afate ; Gh. Xoo. €’AO. '2Ave. -QuecenitAowdtooe“QnfensinefabyA.—JooKasqnceboOsspayemeha @Ued,3A=DQ),vase]dsomegrepG.De, - KeTakwage . _ in©R=besAE“beivarsolgavcepingaligslosWe)de"Vhsasenga WAaLOSE, 1BOD=Aah sersenhsORSQAReCA?Qneke) 3 - \WyreSLA, KOM) 1Wesuds deed»oryEN mg,WaaWEA. ve cdebogt i SXUG.=WAmEA AbYordalonnahboy -Sader: x42)=\2) pasyond ;ERC ACsSa)=e OO aren seterteikYXYKamon . "“QoainiOpnolns” xVea dhWE) eo , 4Deshi Gane =XD=0KVxe6. Sook, Qos women AGS - SQ: DY =okt . : QO) = ante. a PicamdnaDlay Somme. 7 MWegharCrarcauie ot"quadahe! yorwenntog ©doDeWCE) &bedhecer’ han, @Voronue? dhpeMH A&somH. Qewewerk OE)%H=dr) HD.-QA)Ye=SCQye)Me' Wr Os erred’? ae 2h = aomahdbor Te,Kedudgreabgte ath,Wee=MAM) ~AKC... ° . ee oe Po i Hamev mesh | L 1 [see Chapter3\GroupRepresentations. 3:1-Linear vectorspaces. e 3:2. Linear dependence. 3:3. Notion ofbasis vectors inaspace. 3:4.Linear operators, equivalence oftwomatrices (similarity). 3x5. Representation ofagroup bymatrices. Multiplication formula.3-6.Character. i Notion oftworepresentations being eghivalent bysimilarly. Character iseasilyseentobeinvariantundersimilarity, soatherquantity.CalledCHI.Notationis: X%(R)=thecharacterinsome"|forgroupelementR Ky)Kyee++Ky =the classes. Elements ofsame class have same character. 327,-Notion ofdirect sum representation .Blokk form. 3:8.Ideaofafunction beinginvariant under}somegroup(belongs toidentity rep).QM.2:9. Scalar products, unitary matrices.3210.AdjointandUnitaryoperators. | 3x11. Theorem: forfinite groups, youcanalkays arrange allrepresentations tobe unitary.Ie,everyrepresentation iscoupetoaunitaryrep. e 3-12. Definition ofaHilbert Space. 3213. Reduciblitty. Fully reducible means block form. This fisadirect sum. Youkeep looking for invariant subspaces untiltherearenomore,thenyouhavefullyreducedtoirredubibles. |a;=multiplicityofsomerepLareduction. | 3o1k. Shitr Lemmas. ; 1.TwoUIR'sofdifferentdimensionscanhotbeintertwinedbyanymatrix. |2.Theonly matrix which intertwines aUIRwith itself istheidentity. 3:15. Orthogonality Relations. Here based ontheShur lenmas isanexcellent derivation oftheGreat Orthogonality 4 Theorem: SnVIR' | sumB%R),, "4ioON) aySq(9-103) | From this, characger orthogonality isderived) . uses: |fmHR)xR)"=dyy|>canalsowriteasclasssun.et Theorem:numberofUIR'sequalsnumber/ofclasses Lx4 r)Theorem:sumofsquaredofdim'sofULR's=attheorder Zex”ae aeoem 4 po AYaan ae |Ome,xwo. T=de | CCR)=(R))suaeevaileyen — J 4 | -2-| (chapter3,hamermesh) 13-16. Ifarepisasumofreps, thecharacters addintheobvious way. Thus: ® © WeBad@ ay=:cna)ro) a) =K®=ZwK@O a Ke MavKi Yawdeorocker arog dsJam: TCC Pe iA a mabg KOK 1pleatau,ar | This tells youhowmany times aUIRoccurs injthereduction ofanarbitrary rep!!!Whilewerehere,letswritetheothermainformoutsnn — | (F), Ox:=SU . ZV@RW'= Sydybe. | t 1 + | wa! e|sagt g=&Bepro . Nextcomes apotenti little theorem. Takesohearbitrary rep,getitscharacter Xe Then compute the LHS ofthis equation: a (o ha 1. \SZgins Ze G]igavse) Thusy youhave computed thesumofthesquareS ofthemultiplicities ofreps, present inyour original rep. Ifthat sumcomes out 1}your repwas anUIR! Ifthe sumwas 2,youknow your thing reduces into asumof{wodifferent UIR's, each occurs once, and soon.This isacriterion forirreducibilit; Theidea isextended, Iskip fornow. 3217.GroupAlbegrausedtoprovemaincimoree, Heregroupalbegraisdefined,fine.carpeted Theoremisproved:regularrepcontains eachUIRnytimes. Thisisthenused|to confirm thatg=sumofsuxn?of the UIR's, anequality. Next, the group algebra isused toshow that: Ss6)o)*J.OK=(BYTE Thisis“secondorthogonality” te e@ var ,=Gq)%S ‘thecolumnsareorthogonl.Thisalso ©sum ay convirms that thenumber ofUIR'saad equalsthenumberofclasses.“rae | Bat. Progctna ALOK2a push Vad v® Hamermesh, Chapter 5. eComment: Ijust want totrack down afewnew fideas Iseesitting inthis chapter. Sol. Gives the concept ofadirect product representation. This isvery similar toay lectures onthis subject. Inparticular: vier)@)=ofr)ofp) =thedirectproductmatrix You pair the first index ineach pair, and the second indices. Theorem: the character ofthe direct product kepissimply the product ofthe two factor characters, Could not besimpler. Tosee this, just set jl-ikinthe aboveformula andthensum. | 5-2.Thenotionofevenandoddreps.Wawedak. Theorem: ifyoutakeD'xD",iethedirectpoduet oftworepresentations whichare the same, you can doapreliminary reduction lofthe result thho two pieces, each ofwhich may then befurther reducible. ~The jtwobasis functions are: 'S Sex -jis @foypryhe sym ant{isyms. oy osAy ' a * = * RWyssleg SalOyDax|Ss.[Bel») yh aN ne(OwVas>AwWeMl=alWens\. Iwill use Sand Asuperscripts instead ofH]s bracket notation. The idea istosymmetrize and entisynmetrize onthe first~indéces. Your direct product space then breaks into twoinvariantsubspaces.ThebasisraeWau AaS LookSee =ede LAST | dad=AMU) 2) A = - AaNey=Vabewy dain=EMAOYR~1) Itistrivial tocompute the charactér oftHe Sand Arepresentations, just from their /definitions. You get: s) may @*e= tw xe A(py aL Wray nKR ®== NOCOY =CR. -2- (chapter 5,hamermesh) ot - 523, Definition: theadjoint representation gfD(R)is D(R) »called D(R)withbar. e def:thecomplex conjugate repofD(R)|is obviously D(R).= * fact: forunitary reps, Roxx D(R) =D'(R)} sononeed todistinguish them. fact: thevarious theorems involving x(r)” Yes™“th shatbeing x(n!) =E(R),. Thus,thecharacteroftheadjointrepappearnaturallyinvariousstatements. SakeExistenceofhermitianinvariants.a525. Classification ofRepresentations. urThissectionisnewtomeandinterestin}. Ifyoupickaparticular IRofsome group, itmust fall into oneofthree classes} (weareonly talking unitary reps here) * (2)DequivalenttoD”butibemaderealsuchthatD-D™ (1)Dequivalent toD¥andsahbeoughttorealformDeD* /this type ofIRis galled a“real representation" Inboth the above cases, the chpracter isreal for all group elements. (3)DandDYarenotequivalent] (andherecharacter mustbecomplex). eThere isatricky way totell which ofthese tatagories arep falls. You have this: Va ry +t)AKER) =a". ePe tiay& o8) Wigner calls: type(1)="integer" j type (2)=half integer. Example from self: Iwanted tosaythat the|spin-$ matrices involve complex numbers which you can never get rid ofbyequivalencing, even thout the trace (characters) are real. This isa“half—integral rep". On heother hand, the J=l matrices can be made real. This isanintegral rep. On,ly préblem isthat g=infinity for rotation group,butprobably thisclassification can$ecarried oversomehow. Nextcomes abunch ofstrange theorems tboutthings like$(S)which isthe number ofsquare roots ofgroup element S.I]will skip thése for now. There is something here about ambivateht classes which Iskip. 5-6. Clebsch ~Gordon Series forproduct redubtion. Here isthe basic stuff, the character Multiplication idea. Here isthe multiplicity: e@ " »fowea K va Qe=(wey 2 ZX) ® (a =Gyr)5akorn«xxfever -3- : iTheorem: (Proved easily fromthepreceding equation). Consider pDixD y e a)ifusv,theidentity repoccursexactlyonceinthereduction b)ifufv, the identity does not occur atall. Example: InSU(3), the3and3reps arenot dquitalent, but3x3containts theidentity exactlyonce.I:thinkthisisanexarple. 5-7.TheClebschGordonCoefficients. ThisithesectionIhavebeenafterallthis time. The Clebsch symbols are introduced and gfcourse you have toinclude amultiplicity i \sotery 60 Fore ua *Om) an6 Ss = a A seSOTCA [gs) Iftherepresentations areunitgry, then the|Clebsch's areunitary andhave uricus standard properties thatIrecall fromSU(2) analysig,, liketheDD=Dtheorem etc. 528. Simply Reducible Groups i Herewerestrict interest intwoways:[tirst, assume thatallcharacters are real which excludes groups with type-3 reprepentations. second, restrict togroups ewithout multiplicity. SuchgroupsarethenfalledSR.Forsuchgroupswecanprove several interesting theorems. Bythe way,these restrictions apply tothe rotation group, S3andS,. Lemma: indirect products, integrality behaves reasonably. Eg, direct product of two half-integral rpps gives only integral ps, etc. Sublenma: ,D'xD"cancontainonlyintegral (type-1) reps.Thisnotionisofcoursefamiliarfromtherotationgroup. |Theorem: 4faUIRoccurs inD'xD", thenthatUIRmustbelong either toDxD)$or DxD)® ,oneortother. Arepmust beeithe# "even" or"odd", Forreps which dont . appear inssomeD'xD" nostatement maybe fade. Example: Consider 1x1inrotation group. Yopget0+1+2. Thus 0,1,2 areeither even orodd. sincetheyoccurinaD'xB". Since 1inthe]1x1isintegral, thesymmetric onesere the"evens", so0and2areeven, and1is pdd.Consider $x}-0+1. Here, since #is half-integral, thesymuetricwhichis1isfalled"odd".Consistent!.Thehalf-integral |repsdonotoccurinD'xD"andarethisneitherevenoroddinthissense. e5-9. The3-j symbols. Skip. ‘ t Donn igh. Hanmermesh_on Synmetric GroupS,aay. 1 @Jcomment: misisanexcertent reference. AllthosetrickyrulesjuststatedbyLicht. arederived indetail here, though Ihave not}studied it. Chapter is100pages orso. EL,Thissection discusses amethod ofconstfucting character tables forthe§,groups by@sortofbullding-up method. Te,youuseinformation ontheS,,characters topt theS,characters. Ihave not studied the section, butits gaol isclear. Here for example aresomecharacter tables: | ' \\ reas12, S, o @ Sa} OO}: ot @» xe 1 1 An ee ec xo ,oos\ Po a KY2|ofmt! letsgetclearwhatthesetables aredoing. Gincegroupelenents insaneclassalways havesame character, you need only list the classes across the top. For the symmetric groups classes are characterized bytheir cyqlic structure. Thus, inthe S,table aboveyouhave(1,1,1) referring totheidentity, reas(2,1) refers totranspositions ‘and(3)referstofullcyleismadebycombiningtwotranspostiéns. Allthesingle etranspostions are inthe same class, etc. Tha other number tells how many elements are in the class shown. 2-2.ThissectionderivesFrobenius’gvvreformeforcharactersofS,.Bytheway, thephrase “simple character" refers tothe lcharacter ofanirreducible rep, whereas compound character refers the character ofafreducible rep. Here wegetinto allkinds ofpolynomials andthings. Looks ‘horrible. ‘ZB.Butherearethemeumonic results ofqlltheabovework. First,youcandeducethe dimensionality ofeach UIRofS,simply bydounting the"standafd arranglements" of thetableau. Inastandard arrangement, all‘the nnumbers must appear, numbers must increase both across and down. ‘eke Agraphical method for getting the chdracgers. Je. “Characters byrecursion formulas (baded onFreobertius formula) e‘1-8HowtouseFrobenius formulafocomputecharacters. AComment:Allof’tepreceding6sections, {33thickpages,aredevotedtoconstructionofcharacter tables forsymmetric groups! oe ~te Jel. Here isamethod for constructing not the characters but the actual detailed represenation matrices. FollowspaperofYamanouchi 1937.Thesectionendswith t)alarge table which gives thematrices forallrepresentations of51,52, through S,. Here istheS3table, forexample: Ss on TV donne, A eeayoy ey eary (aay as) Fagrey ye 3&wy ela Tay bal [| Details: thematricestare "alwayssymmetric soloneroffdiagonal notstated,youcanfill itinobviously. Also, matrices are only given for transpositions. You can get the matrices for the rest ofthe group elements bymultiplying transposition matrices. By’theway,thesearethesameasthematrices ofTinckhamforBrepofDj.Identify (12)=2A; (23)=B, (13) =Cetc. See page 8ofTinckham. 7-8.Thissection discusses "H{md's Method" ofconstructing wavefunctions 2(1,2...n)@ ofmaximal symmetry orantisymmetry. There isadirect connection between such functions and the Young bisiness. Ihave not studied this and will not because Ithink later section 8aremorerelevant. 29. Group Algebra Approach. This seems tobethe best way ofdoing things. First, ifyou take any linear combination ofgroup operators, you have alinear combination ofh=n!(for S,) operators. Thecoefficients mayberegarded ascomponents ofanh-fold vector. Two such linear combinations may bemultiplied together inan obvious way toget anew vector. Each linear combination isregarded asavector in group albegra space. Pozzi had much tosay about this stuff. Given anirreducible representation ofthe group, you can generate anirreducible representation ofthe group algebra inobvious way. The idea istodecompose the group algebra into adirect sum ofminimal ideals, andthese guys inturn correspond totheUIR's ofthegroup. Itispossible to . constreut projection operators called "idempotents" which take any element xofthe group albegra and project out that portion which belongs toaparticular ideal minimale,Thesearecalled0,,€p...byHammermesh. Ifyoucanfindtheseprojectors, thenthee problem offinding basis functions belonging toUIR's ismostly solved. -3- 7-10. Young Operators. Here isthefundamenthl stuff: consider agiven Young Tableux, ie,ayoung"pattern"or"diagram"whichisfjlledinwithnumbersinsomestandard r)arrengement, allnumbers appéaring once. The|obvious oneisallnumbers inincreasing order, called normal tableau byMessiah. For this particular tableau, you can construct anoperator Sand another one called A. Thest are best defined byMessiah page 1114 andarecalledPandQbyHanmermesh. TheP6rSoperator isaspecial symmetrizer: itisasumofallthe permutations (group elements) which donotmix rows. The symmetrizer issupposed tonotmixrows, theanti issupppsed tonotmixcolums. Example1:8, | B=eeCa)eQd+033) #Cray+(32) Herethere isnothing toentisymnetrize. Note|howthe6elenents ofS,arelabelled here. The lest two are made oftwo transposithons and thus have even "parity". { Example2:s, | :sEl Qee~@)-Y=GP+Gea(az) And here there isnothing tosymnetrize. Example 3:85 nen P=[ere\sonteaoAe(oayT] e+(05)] Kaa Qele-@\\fe- GP]=©-GN-Gs)+(yes). Inthefirst line Ihave added allpossible permutation operators which donotmixthetworows.Thereare12suchoperators.althesum,withparities,ofalloperatorswhich dont mixcolums. i Theoyém: Ithink this isthemain ideay/in o¥der togenerfte the "basis fyfctions" whjéhbelong toaYIRofS,,youfirstlistdffthestandard arrangement. Recall tha! cg) Phereareexactly/as manyoftheseagthereavedimensigns oftheUIR.TMen,foreach standard errangefent, construct P, zhenQ,andthen ofnstruct ¥=QP (this correspgnds totheJobjecfs ofMessiah). This/operator Yjwill bya projector whi¢h willgiveyfo the basis funétion corresponding Aoyour chosen tableau. ~h- Now, here isthemain poop: Youknow that akiven UIRoccurs intheregular gep (ie,thegroupalgebra)Ntimes,whereNisthedimensionality. TheseNoccurances e are, Ithink, correspondent tothe Npossible} standard arrangements you can make inthe pattern ordiagram. Iamnow going tolshow how you find aset ofbasis functions. foreach ofthese Nuirts. (wil... First,eachEReoguugnes. getsitsomprojection operator called¥Y=QP,thej ofMessiah. Youtakeeachprojection operator andapply ittoallpossible groupalgebraelementsxuntilyouobteinastperyfF Example: Consider Se Thetwo-dim repoccur$ twice .Foreach copy wecanconstruct the projector: “ Peeste Yoar= Ann =[e-Gijeroo) a Qee-08) =©-08) f0)~Gar) veHEYA, ! 5 aY=AnSa=VWercattens} =ee)4OS)-Gy=YH. Letsworkfirst withthetopcopy. Takethe projector Yy,andapply itfromtheright (as eMessiah shows) onto various elements ofthe joupalgebra: é na GODY=GdTe~O9 409-(9)=GQ-(usf+e -@) 2Y/ SRTAL, GO =GHYe~ CY+x)-GY}=(D-e +@yce)=-\ : CY =CTE ~Grs 69-2] =GY= GH)+Gu)—@) =YOY . Iamconvinced that ifyougoonandapply ¥,|to allother group elements (thereby spanning thegroup algebra), youwill never gdtanything other than alinear combination ofXYandY's Thus, these maybetaken as“gpanners" ofthefirst copyofthetwo-dim repof83+Infacttheycorrespond exactly t:Lickenbergs PSI1andPSI2. Ifyou want anorthonormal basis, you jist doitinobvious way. Insimilar manner, IcouldtakeXQandg¢nerate aYX,"suchthatTpandYt)"areoperators spanning the other uir's ‘ideal:". . Then finally togetfunctions which corfespong torepresentations, just apply Yyand¥,' tof(x;,x21x3) andyouhaveit! ~5- For example, wehave: AGasy =XYFG)=$03)+Fas)4{Gry-@) ' 1F.2d) =YXFad=FUed)> 48) F3d-Asie) Nowanypermutation acting ononeofthese fictions canonlylinear-conbine thembecause wehavealreadyshownthatY,isabignt-acting projector [thisproves that(12)£,(123) =£11(123), eg]. But,¥,"F(12)Y.Thus GY," =G'¥,=in themanifold. Inotherwords,¥,'isalsoapright-acting projector intothislitleUIR manifoad. Conclusion: Both ofthe above linear combingtions "belong" tothe first copy ofthe 2-dim UIR occuring inthe regular rep decompdsition. Normalization: Since Ifailed tonormalize |PandQabove, ¥,”=constant timesY, s0not quite normalized right. Nobig deal. Conclusion: Inowknowexactlyhowtoconstquctasetofbasisfunctionsforeach @ copy ofeach UIRappearing inaregular-rep decomposition ofS,. Foreach pattern (ie, each UIR) draw thevarious standard aryangements. Each arrangement corresponds toonecopy oftheUIR whose pattern theartangement isin.Then ,foreach arrangement construct theprojector. Apply that projectof fromthe-right-only elements oftheperm group until abasis hasbeen generated. Thenjapply thebais elements toanarbitrary function togetthebasis functions. Ie,afimction f(123....n) . Bythe wey, Ithink the basis functions fordifferent copies ofthe same UIR or : different UIR's are automatically orthogonal} becausse the projectors really are projectors inthis sense. Hammermesh thenwrapsupthissection 7-10tyfinding thesymmetrizers andimplicitly theprojectors forthevarious representaticns ofS,.Fine, nowlets continue to thenextsection. | { .ot =6-i 1 411. Fork, and contruction ofdefinite symmetry wayefunctions. Uptonowwehavebeenconsidering trygbneralstuff,functions £(1,2,3). Suppose ©nowthat this function £(1,2,3.....n) ispealiy aproduct ofndifferent, orthonormal functions. INquantum mechanics, intheindepbndent particle approximation theenergy ofsuchastateisthesumofthesinglestateenergies, andtherefore theenergyisinvariant underpermuations. Youcanthenonlypermutations togetlinearcombinations which belongs totheUIR's ofgroup S,. | Example: n=2, youcanhave u(1)v(2)+u(2)v(1), andtheobher one. Youstart here withtwoindependent functions, andyouregrofp themsothattheyclassify intothe vIR's. Forne3there are6degenerate starting functions. Heclassffies theminto theUIR's. Heonlygivestwofunctions forthe2-dimUI,©soheonlyhasatotelof6functions.IknowhowtomaketheotherreNext, hesays: suppose instead ofallthelittle functions being different, only twodifferent functions were availble. Think Whathappens: yougoahead andconstruct your projectors ¥=QP.When youapply this ¥{to anystarting function, youwill always getzero ifyour tableau hadmore than tworows: reason bsthat youaretrying to antisymmetrize .Ie,suppose youstartwith4tableau havingthreerows.Yousimply cannotmakeanantisymmetric rank-3function ionlytwofunctions areavailble. eIngeneral, ifyouhave kdifferent states only (andnotthefull n),youcan never have any functions belongs toUIR's witl more than krows!TherestofthissectionissomeFockwheIskipfornow.Ithinktheabovething iswhet isimportant here. ‘Tl2:Outerproducts. Ihavetodoasimpler exemple thenhegivestoseewhatisgoing onhere. Consider this: youhave afunction whichissymmetric ,f(1,2) andisafunction forparticles 1,2, Then imagine youhave another "system" consisting ofparticles 3and 4,also described byasymmetric function g(3,h). Thefunction f(1,2) belongs toadefinite representation of85,andsodoes g(3,4)- Nowsupposeyoumiltiply thesefunctions fogether:, Fy(1,2,3,4) =£(1,2)¢(3,4)- Ifyouonly apply (12)or(34)bype operations} this function always goesirito itself end isinthat sense irreducible. But, what hbppens ifyouapply arbitrary group permutations ofthegroup 8,22?Eg,(23)on|this func tiongives youanewfunction. Sohereiswhatyoumight do:Apply elements ofS,tothisfunction togenerate allpossiblenewfunctions.Iwilldotrisonthenextpage: @ : 1 \= 1 sot —Y- ceRiaw= FO93@x) =Bae) eODFla)=£0996,2)=Buy | e (2)Fa(esy) =$3)gGs)=Fliesy) | eR =ts = Flay)| ¢ 903) =fru), ' Cafe =$y)x02) =Gury Furtherapplications ofS,operators willnevdryieldanynewfunctions. Sowenowhaveasetof6functions wietransform amongthenselves underS), ie,wehavea6-dimensionalrepresentationaTheclaimisthatthisthingisinfact reducible, reducible into UIR's of|S).Forexample, ifyoutook thesum ofthese 6functions, you would get afunction belonging to four boxes horizontale. ¥Mem-= mau+ +by. Sothisisanexample ofan“outer produet" representation. . Ingeneral,hereiswhatyoudotomakeénouterproductrep:takeasetof @venpartictes. assumethatyouhaveafunction £(1,2....n) descitbing thefirst r)setofnparticles andthisbelongs tooneof|the UIR's ofS_.Thisfunction has lets sayN-1partners, because this rep ofS,|hes dimension N,This setofNfunctins spans theUIR ofS,- Similarly the second set ofparticles is{described byaset ofMfunctions which span theMdimensional repofS,. Nowmultiply these twosets offunctions together inellpossible ways:younowhaveasetofWNproduct functions. Next,applyell elements ofthelarger group S,, toallofthese product functions. Inthis way you will generate ahuges set offunctions which transform among themselves under groupactionofelementsofS,,..Sincetestare(mn/n)waystochoosenparticlestobelongtothefirstset,Iwouldclaim=“) Cut)a=Sumsusroarlly 2)oulposwg,Hanmermesh thendoesanexamplewhichearleeswithmyanalysisabove.Hethengivestherulesfordecomposing theoutereof+repintoirreducibles. Looksfamilartome. Many examples are given. r)Thisisclear,butfdon'tyetknowthespotutnes (applicebility )ofanoutert)product. Notice that inthe outer product you pantke the "product" ofrepresentations oftwodifferentgroups(possiblythesame)|youthenreducethisproductrelativetothelarger group S.... Very different from inner product. woe 7-13. Inner products. Now wetake twoUIR's ofS, and reduce theresult according toS.What does this mean? Take anNdimensional repofSyandtake itsNbasis functions. AlsotakeanMdimensional repofSs,anditsNbaisfunctions. Multiply e ethese functions together inall possible ways. You now have NMfunctions. Now application ofgroup elements of4S, tothis setoffunctions transforms them amongthemselves. Eg, a We#W=5qnaa: cv)£23) ga(ur,3) 1 x =(Z fesya VSsb;9G) ), i =Zar 2d\q (1s Inthis example, the inner product rephasdimension 20 anditcanbesomehow reduced. Sowhat arethe rules forreducing inner products? . eSpecialCase:Supposeyouwanttocompute (y)x(n-1,1). Thisistosay, e@youwant (¥1:Yp110++) x(n-1,1). Recall that thenumbers inthis notation arejust telling youabouv theYoung patterm: yy boxes infirst row, andso.on.These are partitions ofn,ie,Yyt¥g++++ =nbecause there mustbenboseds ifyouaretalking about arep ofS,. Theanswer isthis: pull aboxoff(y)andstick itback oninallpossible reasonable places. Eachrepyouthereby getiscontainted inthe,innre product. On . topofthat, (y)itself iscontained intheinner product anumber oftimes equal tothenumber ofdifferent values ofthevyless one. Example: Sa oy) 2)m*B =&+QDgecko) =ray OF aBeBe MBs = e Oy A)ootDaackatWAsD=[2 deine -8- This last result follows since £(1,2) g(1,2){is obviously 12symmetric ifboth functions areseperately. Thus,thetotallyspnmetric representation ofSislike e the singlet ofrotation group orany SU(n) grpup. Notice that these results tell you: “the] product ofasymmetric function with anantisymmetric function mistbeantisymmetric" andsoon. 7 1 Example 53}, Cane3 Bea-o0 Jdonee Racraiyfaster)mare(ns) akomyamsgosby ne.aatycoped ' - CoatyCass.0) . TE) EY =EIT [yuote 1 Cane31 Aer=8end[odeSonWe)feQi)=TH ForanyS,group these three products arealfays obvious: total symmetry andtotal antisymmetry. The difficult stuff iswhat hapbens’ with the other reps: (Coan2) °Ty FS=PRsGye) = io) ay) Crer. |wot .-Saat = fey w= ay A+ (Pot) =mors ay Gy urvele -Cone3 Be P=Pe(@\b Ro. prem { Andthete youhaveit:Ifinally seewhére Feynman gotthose facts (eg,mixed times mixed=S+A+Metc. Hanmermesh gives someuseful general fordulas andhasatable forS,andS.. Te1k. Now comes_the Clbbsh problem: for example, wesee above how the inner produc oftwo(2)1)repsofS3canbereducedinto42+(241)+(13).Whatwedonotyetknéwis'whichlinearcombination ofthe4functions yougetbymultiplying thetwo @starting repfunctions belongtowhichrepsinlthereduction. Ie,wedon'tyet.know the Clebscho Gordon coefficients! | |; Aside onYananouthi Symbols. Recall that aUIRofS,ofdimension Nhasifso-called standard arrangements. Wesort ofidentify each standerd arrangement with arowoftheNdimensional representation matrix. TheYamasymbolcanberegarded asalineare way’todescribe the standard arrangement. Eabier towrite theYsymbol then todraw the piture everytime. Example: Weld aaaist (esa ch Howdid Idothis? Look forthe highest number, the 6,Itisinrow 1ofthe tebleau, soputa1asfirst symbol in[ J.Where isthe 5?Row 2,sonext symbol isa2.And 50 on. Fact: ifyou read the symbol inreverse, itisalways a“lattice permutation", ie, there are always more 1's than 2's, more 2's than 3's and soon. Iguess this can beshown tobetrue, given the rules for making the standard arrangement: increase tothe right and down! Now back tocomputing the Clebsch's. Hanmermesh gets very general and complicated recursion relations andsoon,buthealsodoessomespecial cases. @ Case 1:When youcombine arepwith itself, theidentity (totally symmetric thing) occurs once inthe result. TheGlebsh forthat combination isvery simple. Example: an Oy @® al’ Px =m+ fP+F L}—l 59i)GYany z = eck & LuarmaypoleingWes. Oxa4Giva Bia\Wrp 7 oo“ 'sane Sobek[nerma FTO =PPG, 1@,Lad)Greaye \e +[eqtrg’? a(Pet=pin mM=-luyy: ®~¢sfirg Notice that the Y-symbol actually includes information about which rep you have: i8, allbyitself ittelys youwhich UIRandwhich standard arrangement (row). Myresults hereagreewiththefabletheygivelater. @ : \ 1\e Ot) = w\RYLe, Yonds FZ wig -~9-Let,me:repeat:thislast.example: |vr. Wi+zitaytany toatindl a ae an oe ca: {PR+eel ‘This tells you exactly what Linear combination of"basis objects" goes with the totally symmetric rep. Ithink itwould beuseful to deeexantly what this means. Each ofthese M(mixed) representations has two basis functdons. Then here isthe claim: a . Fars=xeSau3.(ye)+£desygerng Qe)FAS)FOB)|ote|. Itcertainly isnot obvious tomethat the function socon structed is‘totally symmetric. However, wecanshowthatdtreally workdesfollows: consider conbining an earbitraryrepwithitself,andyouare="theidentityintheproduct.Youhave: OXX , 1 leq=e 2S, 5. DoD a pva\eyal 1SSC} apy & we = vy Crcanstan? Rios Fe AMoeRog » — 2 ,=E¢Di(e\ [rs =Di®DY =\ » ay ‘* &De@PDe @DADS BEADCADLE)=Dale~3g1ne|LApeelTAyRysi>[-weaSUA+To. SonowTbelfeve it.Thisis«goodClebsh sgttoknowbecause youareoftencombining twolikeobjects toseewhatcombination is4tallysymmetric orwhatever. Case 2: When you combine aUIR with the transposed UIR, among other things you get thetotally antisymreponce,the(1")rep.HerearetheClebsh's forthiscase: \ tina) .,® oma erent : LO FAW Acey Here the parity isjust the number oftranspositions away from the standard normal arrangement. Here isasimple example: . a 1 aIeG)Te)eI . bays lw) ti). Here you are combining two‘one-dim reps, sonot much todo. Since only one possible arrangement ineach, parity is+1+ Another example: e : ue : coke(ht(R=Dt)aOe Rx fP= +a eek. =+ ‘ 4t a,PP+OPWe x ' a kAtye ALbyten- Teyfa? Jequacate « ® -10-| Case3:GeneralRule:whenyoucombineany|repwiththeTArepyougetthetranspose e ofthe original rep and nothing else! Inth4s case wealso know the clebshces. Me. aBa=AbALOXAL, | From PxG- Bangle* ‘ q*A*W _B-BE ' 7p-cpy pom: Oe : a @mw:A=> MY .3||. Conclusion: forS31almost alltheClébshes drecovered bytheabove special cases, seelabelling inred.ThecasemissedisalsoFiverinatablewhichIwillquotebelow.Note,bytheway,thattheClebshc's orebasedonadefinite convention for therepresentation matrices. Inotherwords,pereallyhavetocoordinate your representation tables with your Clebsch table:OnthenextpageIgiveacompleteS,cicbshtable: (seenextpage)Bytheway,Ithinkthereis|symmetrypropertywhichsaysit doesnot matter which order you putyour twoobjects. End_ OfHammermesh Chapter 7onSymmetric atte -\I-GompleteClebschTableforS,(seeselsfor2-dimrepmatrices) a m8 [mB] | ee feof: fe a ear |§P|ge|efe|sfi eeiee SE |ee lee (ep|pe] mr |-[xtiale | Problem: Howcould one compute the Clebsch's from scratch for thig group? @ +:Firstetesindouthowtomakeatotally bymmetric combination sincethisisthe easiest todo.Weknowthereissomelinear cbnbination thatworks, callitV,t = | Ng=ES RG Q) dheFeBGeBR 1 Rix=VeVR@) | Ifweinsertequation (1)intoequation (2)as.applygroupoperation RtotheMbasisfunctions weget this: Ss 48 &SyVUMDBINKGE =|2Q HG; qe <8 sy eS ny : s 8=Ge2 mOE) ve:% . s so jy ee.f= DED ;>* D=DB=H i t = Ss =FIW=WS | Iexamined thisinsomedetail,butnowIseepfastwey:C,isamatrixwhichconmteswithallthematrices ofaUIR.Shurt tells us|thatC°ispooportional totheidentity. Thus wemay conclude that: , Ss8 s - Cy=KS Ks amahendd, Cy Tofigure outtheconstant, wecanapply thecompletély general formula (5-111) which OMtury \Cudye ZAes VEYNGEVYIMBS) =SpyBugSee Forourapplication here uev= themixed rep, there arenomiltiplicitly labels needed 80we have ashorter version: | :Xs! ds \* e ZL.CHIC y=Sale o whereheresistherowindexfortherepyouf"combininginto".Forourpresent caseweinsertourresultinthediagonal=pof(3)toget:dee cs u @Sisl-t se KrZlayy=1. s &SpatL et. GZyye Ze zie 2pKiag AkePe This phase ischosen zero and wehave duplicated the results ofHam's table. 2.Comment: Notice that this family ofClebsth coefficients into the Srep does notchange signunderinterchange ofiandSze, wehave: (SIMA) =CS1Wj,Mi), \ According tolaterdiscussion inHanermesh, I]thinkthiepeansthattherepSis contained inthe symmetrized subspace ofthe product D! and its delta=+l. For thegroupSyweknowthatallrepsaretype(1),realrepresentations, ie,theyare all integral representations inWigner sense.{Thus, rep $isan"even" rep. 3.NowletslookforClebshcs intotheArep!| SitYonmonyYocdong|heansHea» (ey) ~ Sie @ah-2Viot ~2Octre dw =ACyDec(ODesCYRGetAyre N ay wm =>|DanQ@®Cre==VyDelODBR*) |(4)t ‘This appears as(5-114) Hamermesh. This istHetype ofrelation that allows you to compute any Clebsch's. Along with the normal: zing consitions which I}1l record here: Sot attated (5)ryCyC5=Serbanyen H 4Sem Om 2Syded &)¥ BMY Brey -3-Nowrewriteinmatrixnotation,thenoltotheA-rep: e a wt |YoLTZVaCes ZDYOCTDEM ‘Sy nN yt law -\ >ZnO ILT=RATES] WreA, OeDee Sedu |oe A <\ Be(I= Dele Wow. ! GeCa\(OsGre =(OwOry(ey& .(SGSS enanNeek) Because ofthat delta, Inolonger have a"Shpr" bet, sohave tolook atvarious R's eandseewhatweget. GWC(t © o\ (ey© Qe-& aons(ANG EIGS) >Ske Qua, atech=o. Qua: Cady CaDee. Yer WSeeCaa|Ls Dacl ar@=De. | Cay SDa J4 |Ae) = see He,| GQ,Ly LL Heo DrCen)=B=Seay oh kr°o1A atonAt e~ ee Le :-Go)¥ :. AAA .arty (fie Ate Nomsea: in(crey=\5WAG) =oe -a4 3.Nowletsgetthesephasesright.SupposefchoosetheS-repphasetobezero. @ ment amprobably not free toBoose this phabe also because wehave acondition: Yetdont (aIe"TsBaBe)(9) sAah | ar lESX|=o Bey + 3G SsAo A1eMA/o re onl teC-kG) Ge "Gs gh-aSe 7G?) soft BkOrSA apectuce JnergSe.AndesSS, IthinkthateachfamilyofCcoefficients calhaveitsownarbitraryoverallphase (the same forallmifthere isanmrow labe}). You can seebyinspection that equation (4), the primal one, isinvariant under anoverall phase change ofthe Cfamilies. The same applies to(5)and (6) dince the*appears there. SincetheD'sareallreal,youknowyoujcanmaketheC'sallreal.Butthere @ isstill anarbitrary overall +1 for each Cfamily. SoIwill choose toagree with Hamermesh: Ot Aja AL Gh.+eGe) alCwye Qe 2CAsmimyy= =CA\Ay Kd Therefore wefind that the A-rep isan"odd" xepresention. 4.Nowcomesthetoughie.Letbethemixedhep.Thenyouhave:i : i " ™ JDan(®) UWE) -QU) os CuzCa=o * Yar ReQe). Quan yal 1 \ ¢Ch=).\) Vyryo1s Z T = v ve (ei =GAYLE] Ga[>CeFNS)—Cae/ .oy =(2Ts"\ ah aN Sothat was rather helpful. Now lets gotoR=13): eAVW[Cl= WOC+GaceLC] =-1)e')-8rd*) -“cy|BCe SyfootsohCh)beet: |\ RACY Bar) 1& -\S\/G& 2chAER, 8)-cafes) &) ‘ a \ u5Reals eG LxIcaGe|=-2Bea sYaleysce]=fcHeGch)=HER) O43=220)ects-feek tLees =-2¢) 3Cy-Ca=+2Ch 3Gsdeo alee) [ead axe +73Jen-s) meae-tt eed, Br Crea. Oa \ a 0 a o784= ond =y=(B=) (cCo) Sowéhave reduced ittoasingle parameter, a Nowapply equation 7inallways toseewiat gives: algighye afalCtr taGeb, kISPV=CAcata(CEGs}s (=teasone @idtheoy 2K(sayhijeo=-4 KC} oY ros 6.| Soletsconvention thata=+1/SQRT2.Then|have: vt 7 \ @Qu:k Que-i\ 2 :es -#Cus Ca= HE This agrees exactly with the table. TheMrepmust beeven because switching thelabels 12does notchange sign. Te, Ci(Ig) =FCth GRC) wmahar Conclusion: 1)Thavederived llHanermesh's coefficientd forS, 2)Ineach family, anoverall phase canbeadded. Don't doit. 3)TheSandMrepsare"even" having delta =|+, whereas theArepis"odd", de~. Thesignificance isthis: suppose youaredofng MxMinto M,S, orA.Ifyouswitch your initial labels, thecoefficient will charge sign ifitisgoing into anoddrep. Youcancheckthatthisruleworksforthewhdlesubtable. e@ 4)Ifyouswitch inthissense inanyofthe dthersubtables, yougetaplussign. Thisisageneral symmetry rulederived byHinlaterchapter, something Icould probably show but amtoo lazy now. i . EEE an | Toe), 19i Hamermesh, ‘Chapter 10:Irreducible Tensofs baallthat. . e10-1.ThemostgenerallineargroupofsdunisofcourseGL(n),.Allgroups ofinterest are subgroups ofGL(n), soanythifg you can say ingéneral about 6L(n) ‘is very nice. Thefirst thing youcandoisdefine a"Yector" with respect toGL(n), x,,having ofcourse ncomponents. InSU(3), eg, a"vectér£ therefore has3-components. Ofcourse there aretwoinequivalent 3-dim reps ofsu(B) sothere are really twokinds ofvectors, but consider either kind. Next, atensor ofrank rwith respect tolGL(n) ,orwith respect tosome n-dim matrix group, isdefined. Just atransformatign law. One obvious way tomake’such a tensorisFuggoeax?) 1d,youcancombiner different vectorsgustmultiplied together. Notice“that there afesubindices ontheindices. Thistensor has ntimes rindependent components ifall yectors are different. 10-2. Decomposition ofspace ofrank-r tensor into invariant subspaces. Gonsider somerank-4 tensor, seyGy145); »PikoneoftheYoungsymnetrizersforthe,group S,andapplyit-tothe ++subindices ofthistensor. Inthat wayyouprnmuxc produce @tensor having adefinite symmetry wrtgroup S). NwtakethisnewtensorandthinksboutHt.Bg,youmighthavethis: e@2a tem A .1 # “This ‘tensor isassociated withaparticular rowofaparticuler representation.etofgroupS,.WhathappensifyoulapplyageneralGL(n)transformation “Yothistensor?| * | 1 Fx=Diytig,BiniBiayhelis| .. i . RS =AWApayy, ag,\VomiGo. . AyeeQuis Qingy Sivysyy 4,-TipsGoniiy Youseethat the"rotated" tensor hasthesamd symmetry classification oftheorigianl tensor. Any tensor canbeprojected into irreducible components which belong toUIR rowsofUIR'sofS,whereristhetensorrank..Ofcoursethereasonthisistrue e isthat theGL(n) orwhatever group "transformations" commute with thepermutation group action ofthe Sgroup. This isimportant end isworth some time. Arank-r tensor can beacted upon by group operators intwodifferent senses: elements ofS.cenpermite thesubindices,candelements ofsayGL(n)canactonthetensorintheusualsense.Thereasontheseé actions commute isrether obvious, eg, ’ CHR=GOTRa=GdSawai Ryvo =zAasAeRae==4GayePuy ‘ =zAeFie(OY)Fd=TeOdFra Que, [Tad =O|comme Now, isthere any connection between the rank rand the vector size n? Not really, except this: inanyS,tableua there aregoing tobeacertain number of rows. Thig number isofcourse some number less than orequal tor.For that tableau with Rrows, there will exist notensors atall ifnislarger than R.Reason: infirstcolum.yopmstputRnumbers, butat,least.onenumbermustberepeated andthise kills you under antisymmetrization: Remember that Y=QP sothe entisym isdone last. Inparticular, ifnisgreater than ryou get notensors. Another, way tosay it: ifyou are given avalue ofn,then the only tensors you can have correspond toYoung diagrams with less than orequal tonrows. Once more: suppose someone asks you: What irreducible tensors can you make wrt group SU(3)? Answer: you canmake tensors ofanyrank ryou want. For given rank r,the tensor isclassified byYoung diagram with rboxes but cannot have more than three hmurmcknx rows inYoung pattern. Bytheway, atthis point, “irreducible tensor" means atensor ofsome rank r which belongs toaparticular UIRofS,. Sofarwehave said nothing about representations ofthe group GL(n). Next, Hshows how tomake TSorTAtensors out ofvectors. Itisvery easy to make aTStensor ofrank-r. All you need isone vector x,then multiply rofthem togehter and you are done. Tomake anTArank-r tensor, you need rdifferent vectors. The TAtensor isthen just’ the Slater determinant. Ofcourse ifyou are making TAtensors, youcannotmakeonewithrankrgreater thann,asdiscussed above.Ifrenthetensor eyou get 18maximal insome sense and isproportional tothe epsilon tensor.!! -2- 10:3, UIR's ofGL(n) ,their dimensionality. te WEknowthatwhentheGL(n)groupactsoharank-rtensor,thecomponents ofthis etensorgetmixedaround.In particular, suppose wehaveanirreducible rank-rtensor. Weknow that after action byGL(n) operator, tensor keeps itssymmetry relative to Sp Letsgettothepoint. Suppose youhave4rénk-r tensor andyouhaveGL(n). This tensor then hasrxn components which are mixed among themselves byGL(n). Thus, ‘thecomponents, ofthis tensor form amxrdimensional representation ofGL(n). But such arepresentation ingeneral isreducible} - Hereisthedeal:‘nowtakearank-rtengorwhichbelongstosomeUIRofS,.Because thistensor hascertain symmetry properties’ (dg,might betotally symmetric), notall nxrelements are"independent" (ie,they ar§notlinearly independent; eg,perhaps Fig=Fo13)+ Suppose there arereally onlyN|inddpdnenet elements. Thenclearly these Nelements aregetting mixedamongthemselves joyGL(n). Thus,thisirreducible tensoris.givingyouarepresentation ofGL(n)ofsift NjOfcourseitisalsoin arepresentation ofSofsome other dimensidn. Weneed anexample: Consider arank-3 tensor belonging to(3)rep(ie,theTS).UndertheactionofelementsofS3,thisaitransformsas2onedimensionalrepofS,__ regardlessofthevalueofn,But,itturnsontthatthenumberofindependent e@ elements ofthis tensor is(n+2/3) binomial poefficient. Ifn«2, this tensor transforms asafour-dimensional rep ofGL(n).! Recall that theelements ofS;here actohly on‘thesubindices; thatistosay, elements ofS,actonthepositions ofindices. Whenyouarecounting thedimensionality oftheS,representation, youarecounting Linbar combinations ofthethreeobjects(three positions, orthree subindices) which, thansSérm among themselves. Ontheother hand,whenyouarecounting. theems theGL(n)UIR,youareinterestedinhowmany,Values anindexcantake. . Sohere isthebigquestion: forGL(n) forsome fixed n,andforatensor of fixed rank.r, andforaparticular Young diagram with rboxes, exactly howmany components ofthistensor areindependent? Howdoyoucont theindependent components ofa tensor ofrank rwhich isirreducible with respect toGL(n) according tosome Young @iagram?? Answer: youcount the“standard ardangenents". When wecompute thedimensionalityofaUIRofS,whenalsoohoarrangéments,butinthatcase every integer from 1tormust appear oned. Inthis case, only numbers inthe range 1toncanappear (nmaybesmaller orJarger than r),andmost importantly, numberscanappearmorethanonte,aislongastheygivea"stendard arrangment" with enumbers’ thesameorincreasing across, increasihg down. Thad,better doanexample ‘tounderstand theorigin ofthis counting rule, other wise I'll forget itquickly. Example: Consider arank-3 irreducible tensor belonging totheMrepof33.Lets count the components ofthis tensorf Tes|Fp--~|fip-fp-fip- fp-° |Reab.Fe_-b| So,ingeneral atensor ofthisformwould haven?=2?=8components. Butforthis particular case you seethat 4elements vanish, and the other four arepairwise related, soreally there are only two independent components. Idon't really understand this counting stuff inmore gmplicated cases, but I accept the result. Icould figure itout ifIwanted toinvest the time, but right now Tdo not. But Iknow the rule: just count the standard arrangements. Ialso know ashortcut tothe rule which Icall the a/bmethod. This isafast method ofcounting the standard arrangments. Here itis: aa fei on zEE ges ts=>3 Saogee, 48 e Ilearned this shortcut in,Lichtenberg. So, byexamining irreducible tensors and counting their indpendent elements, wearereally counting thedimensionality ofUIR's ofGL(n). Outer Product Analysis. Now, observe that a"vector" isarank-1 tensor andit belongs tothe[)representation ofgroupl S,,rather trivial. Nowsuppose youwant tocombine twodifferent vectors (two systems) gomake a rank-2 tensor. This isprecisly what that outer product deal did foryou!!! Recall that youcombined anS,with an§,andclassified results into S,._. Soconsider: Den-= m+A-\ XeYS KYA +HHS Y]ieee eeHOt ” Here wearecombining tworank-1 tensors togetsome rank-2 = tensors. BUT, eachirreducible tensor isassociated with eUIR.™~ ofGL(n), sothispicture alsodoesreduction ofproducts e ofGL(n) UIR's, .(Ofcourse inthat sense itisaninner product since each diagram refers toarep of6L(n).) wee -| Example 2:Hereissomething interesting. itrank-3 tensors canyoumakeby ecombining asymmetric rank-2tensorwitha“peer? Men= On+ a) NowIwillbeveryexplicit withanexample: |g.,plusamomtnum, nelLorenta. Yn,=CaP om . Yama Syste=F LgphGa+SagtiSayP| This istheonly completely symmetric tensor Youcanmake from these objects. Next: XB=TAS AELA GILQuedas 2dpePy = >SLapmPad—Qefa] : Yeo SDHCOMON,Gayefs=[FOOD] (LyreBog+Joyefn) * a=3Lopypefa=vent . “Tn=3Lapp PeeSikPatsayspa] ~2 Tp =FLaeens-amydpa- q / Te st *Spya—SymePr| ; To Sp Notice thatthe"other" combination ey -@Pu3isalinear combination ofthe .twoMtensorsshown,itisnotlinearly{esosSo,wehaveconstructedallthe e tensors having anyS3symmetry that itispossible toconstruct fomthese objects!! And theyaddupright.ObviouslythereisnoinTAcontributionhere. 10-k. About thesubgroups ofGL(n). Thepoint here isthis: ifyourestreit yourself tocertain subgroups ofGL(n),theUIR'syoufound(andcounted) inthepreceding sections stillapply.Ie,ifyoue have aVIRofGL(n) andyourestrict ittoGL(n)' (real only, GL(n,R) ifyoulike), the rep isstill irreducible. That might not have been the case. Here isapicture shoing the subgroups where irreduciblity ismaintained: aArt L rT . —s Thus, theabove analysis gives thedimensionalities ofthereps ofSU(n), forexample. Inpassing, Hhasafewcomments about Lie Algebras for some ofthese groups GL(n) Lie algebra: 2{n) Kg=Bg vdbetsdemanle. (28padgoreAD A D5,Kaw)=Baye SonKh LieAt-Xy=3; : Qu) U(n)LieAlgebra: @Yay=%@Sabh.+ body UJn) . . a .KO.Bide —CSeHLAhasteecon, ky Aoy y ws)Xy=53,3 Fact: ifyou take any Young tableua which stands for arep ofone ofthese groups, and ifyou add 2colum ofnboxes, the claims isthat you make nosignificant change totherepresentiation: youcauses ittohave extra factor (deta)® carried along ifyou add scolumns ofnboxes. Thus, essentially you ene free toalways cross out"full colums". Thus, theUIR's ofGL(n) orone ofthe above subgroups are really enumerated byYoung diagrams having less than nrows. 10-5+thecaseofO(n).Nowsomethinganteneydifferenthappens:here,whehyou centract atensorontwoindices(takeits"tface"onthatpairofindieces) the eobject you get transforms asatensor ofrank|2 lower than the original tensor. In theGL(n) case youcould still dothis contragtion, buttheobject yougetwould nottransform asatensor: inO(n)youhavethatextra property oftheaj5which makesthiswork:nefiely,ajaiyndye +*”| boa, :Sowhat,isthesignificance ofthistact? Supposeyoutake@tensorFy4, whichbelongs tosome synmetry of3.Youmiglt think that this tensor has independent components and these are shuffled around inaninvariant subspacebytheO(n)operationsandthissubspaceisasrtnotso!Thereasonis that this tensor, orthis invariant subapace, [actually isreducible further into twopieces. ConsiderL: Fouts=(Fens) +(daaHa+SaLaetiKi) mhTheclaimisthatanytensorcanbebrokenwatwopiecesassuggestedhere:thefirst pieceiscompletely traceless (ie,contract ohanypairofindices andyougetzero). The second piece. isthe remainder. Youcanshowthateachofthesepiecestransforms onlyintoitselfunder0(n) etransformations, thé reason being that "contraption" operation, like the symmetries, commutes with theO(n) transformastions. You cancompute H,L,K the three vectors appearing inthe above just bytaking traces. The usual simplest case ithis: “> Ra=RatSav=F3RasfatGaBi =oRas [hae Ba cyBE Hereyouseethegeneralidea:firstyouconhe‘thepiecewiththetrace,then subtract tofind the traceless pensor. Anytensorcanbedecompoesed inthisway}Intotraceless plus4,)tinesLover rank tensors. 10-6. Nowconsider thetraceless tensors. The|space oftraceless tensors isinvariant underS_,permutations! SoifyouapplyaYoungsymmetrizertotracelesstensors, e@you generate irreducible tensors wrt O(n). { Theproblem now forO(n) istocount thedimensionality oftensors ofvarious S,,symmetry which eretraceless, Obviously this extra tracelessness condition is goingtolowerthenumberofindepéndent components. r) Example: View Fin,FayFae,Rae Bn6t(2) Fa,Poety /4. O@). Here the trace conditions easily give you the second pair ofelements interms of the first, hence only 2independent elements. Before welearn thegeneral rule forcounting inO(n), here isatheorem: Theorem: ifyour Young diagram ofS,forarank rtensor hasmore then nboxes in thesumofthefirst twocolums, then thecorresponding traceless tensor vanishes identically, ie, has zero independent elements. ome Fam =O ateZeBt 4x6h}.©), Thus,‘yousimplycannotmakeatraceless rank-3tensorrelative to0(2)whichhas e this symmetry! NowHgoes ontogetcounting rules forspecial cases like 0(3) arid0(5). Basically you just count how many conditions tracelessness implies, ther subtract those off the count for GL(n). Next Hshows howO(n)* isspecial case ofO(n) 10-7. Now recondider thedecomposition oftensor =traceless tnesor +extra piece. Thetraceless tensor with S,repchoise forms anirreducible ¥pofO(n). Butthe extra piece canbefurther reduced. Write second piece as(double traceless tneosr +residuel’ lower piece having twodj,functions andsoon. ‘Thepoint ofdoing this issimply that youareshoiring howto"reduce" aGL(n) representation into O(n) representations: Idont care nowabout this. 10-8,10-9. Symplectic groups. Skipped. .