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Old App A stuff
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Working notes in Phil's tensor wedge project, kept among old versions of sections removed from Appendix A. They give an older proof that a signed sum over permutations of products of Kronecker deltas is unchanged when P is replaced by its inverse, using the equal sign of P and P-1. They also include a draft section on determinant forms using the Levi-Civita symbol and permutation sums, with row and column versions. Dated about January 1999 by the notes' own comments, though the year is unclear.
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Old proof of the raise lower P on delta theorem PhL 1.5.15
I removed this from Appendix A because I found an easier proof as part of Section A.7
1.15.16 There is nothing about this in A.7 now! I think the NEED for this theorem went away.
A.6 Theorem on raising and lowering P
ΣP (-1)S(P) δP(j)i δP(j)i... δP(j)i = ΣP (-1)S(P) δjP(i) δjP(i)...δjP(i) (A.6.1)
Define Q ≡ P-1 which is the inverse of the permutation P of the k integers {1,2...k}. Then,
δP(j)i = 1 P(j1) = i1 j1 = P-1(i1) = Q(i1) δjQ(i) = 1 (A.6.2)
so that
δP(j)i = δjQ(i) . (A.6.3)
Then
ΣP (-1)S(P) δP(j)i δP(j)i... δP(j)i = ΣP (-1)S(P) δjQ(i) δjQ(i)...δjQ(i) . (A.6.4)
But it is a general fact that, if Q = P-1, then
ΣP (-1)S(P) f(Qx) = ΣQ (-1)S(Q) f(Qx) , (A.6.5)
because S(P) = S(P-1) = number of swaps in either P or P-1, and summing over all P is the same as summing over all inverse permutations P-1 (either sum exhausts the permutation group). Therefore, replacing dummy Q by P on the right,
ΣP (-1)S(P) f(Qx) = ΣP (-1)S(P) f(Px) (A.6.6)
so the right side of (A.6.4) gets Q replaced by P so that
ΣP (-1)S(P) δP(j)i δP(j)i... δP(j)i = ΣP (-1)S(P) δjP(i) δjP(i)...δjP(i)
which is the claim of this theorem. QED
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A.6 Some Facts about Determinants
A standard form of the determinant of a kxk matrix M is the following (see *** )
det(M) = Σii...i εii...iM1iM2i ...Mki
= ΣP (-1)S(P) M1P(1)M2P(2) ...MkP(k)
= [ M11M22....Mkk + signed permutations of the second index ] (A.6.1)
It is easy to show from the first line that if we swap any pair of "first indices" on two of the factors in M1iM2i ...Mki, the determinant changes sign. This is just the well-known rule that a determinant changes sign if any two rows are swapped.
Because of this fact, we may write
εjj...jdet(M) = Σii...i εii...iMjiMji ...Mji
= ΣP (-1)S(P)MjP(1)MjP(2) ...MjP(k) . (A.6.2)
If two jr are the same, both sides vanish (determinant has two identical rows). Otherwise, the εjj...j on the left implements the fact just stated above about swapping "first indices" on the M factors.
If M happens to be a covariant "up-tilt" matrix, then (A.7.1
εjj...j det(M) = Σii...i εii...iMjiMji ...Mji
= ΣP (-1)S(P)MjP(1)MjP(2)...MjP(k) (A.6.3)
Since det(M) = det(MT), we can write (A.7.1) as
εjj...jdet(M) = Σii...i εii...iMijMij ...Mij
= ΣP (-1)S(P)MP(1)jMP(2)j ...MP(k)j . (A.6.4)
Again, if M is an uptilt matrix, this reads
εjj...jdet(M) = Σii...i εii...iMijMij ...Mij
= ΣP (-1)S(P)MP(1)jMP(2)j ...MP(k)j . (A.6.5)