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# Class Note for MATH 796 at KU 7

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This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Kansas taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

Monday 428 Omega Last time7 we saw broadly how to use triangularity arguments to show that e7 3A and pk are bases for the ring A of symmetric functions the rst two Z bases7 the second two Qbases Triangularity does not work for the basis ILA because the complete homogeneous symmetric functions have so many terms For example7 in degree 37 hg 1 1 7213 H 11 31lm211 h111 1 6 mm and it is not obvious that the basechange matrix has eterminant 1 although it does We need a new tool to prove that ILA is a Z basisi CADIOH CL De ne a ring endomorphism w z A A A by wei hi for all i7 so that we h This is wellde ned since the elementary symmetric functions are algebraically independent recall that A E Re17 e27 i i Proposition 1 f for all f E A In particular the map w is a ring automorphism Proof Recall the generating functions 1 Et Zektk H1tzn kgo n21 2 Ht thtk Hail kgo n21 Using the sum formulas in 1 and 2 gives 3 EtH7t Ziektkhnkitn k 2t Zn71 kekhnki n20 k0 n20 k0 On the other hand7 the product formulas in 1 and 2 say that EtH7t 1i Equating coef cients of t gives 4 271 kekhnk 0 0m 21 160 Applying m we nd that 0 71 ikwekwhnk n a o 71 7khkwhnk H wMS SH 0 Ukllnekmhk H o n 1 ZEDthnithk k0 and comparing this last expression with 4 gives whk eki D Corollary 2 hA is a graded Zbasis for A Moreover AR 2 Rh1h2i i By the way the equation 4 can be used recursively to express the ekls as integer polynomials in the hkls and Vice versai A Bunch of Identities The Cauchy kernel is the formal power series 9 H 17 ziyjfli 13121 As welll see the Cauchy kernel can be expanded in many different ways in terms of symmetric functions in the variable sets and For a partition A b n let mi be the number of is in A and de ne 2A lm1m1l2m2m2l 5A ilm2m4quot 7 For example if A 332lll then 2A 133l211l322l 216 The notation comes from the fact that this is the size of the centralizer of a permutation a 6 6 with cycleshape A that is the group of permutations that commute with 0 Meanwhile 5A is just the sign of a permutation with cycleshape A Proposition 3 We have the identities 5 H 141ml thmmy EM 1321 A A 2A lt6 H Hm Zammmy 2am 27 13121 A A A where the sums run over all partitions A Proof For the rst identity in 5 H liziyjfl H Haimrl 121 1321 13921 g H 13921 kzo m lt7 H Emmy 121 1620 Zh m y A since the coef cient on the monomial ylflygg in 7 is hmth For the second identity in 5 we need some more trickeryi Recall that 3 14 42 q l 1 1 i if 7 0g 4 was n q 2 3 Therefore log H l iziyjfl lOg H 1 Iiyj i E 10g1 Iiyj 13121 13121 L121 1 ll 1 n n ZZTy Zgzxiyj ngingi n21 13121 W and S exp anzfnygt n21 l n z n lt gt5 m 7 i k P1IP1y m1 P2IP2y m2 QMLQAM 1 H 2 gt J 7 PAIPA9 2 i D The proofs of the identities in 6 are analogous and left to the reader Corollary 4 We have s hn 9 AH 2A 9 en 51 and A Min 10 wp Sip Proof For 8 we start with the identity of 5 z thzmiy 2 10 if y x A Set yl t and yk 0 for all k gt 1 This kills all terms on the left side for which A has more than one part so we get WW 2 mm 2L 2A An A and extracting the coefficient of t gives Starting with 6 and doing the same thing yields As Brian pointed out you can t obtain 10 just by applying w to 8 and comparing with 9 as I had mistakenly claimed in class Here is a better reason In what follows w is going to act on the zils while leaving the yfs alonei Using 5 and 6 we obtain 2 Hikipmy Zh m y w ltZ AImkygt w Egkwp if yv A A A A A 2A and equating coefficients of p y2 as shown yields the desired result D The Hall Inner Product De nition 1 The Hall inner product lt on A is de ned by declaring hi and mm to be dual bases lthA7mMgt 6AM 0 Two bases ui vi are dual under the Hall inner product if and only if 1 Z uXU A L121 1 7 Ii 0 In particular l A b n is an orthonormal basis for AR so lt is an inner product 7 that is a 2A nondegenerate bilinear formi o The involution w is an isometry iiei lta 12gt ltwawbgti It sure would be nice to have an orthonormal basis for Ag In fact the Schur functions are such a thing The proof of this statement requires a marvelous combinatorial tool called the RSK correspondence for Robinson Schensted and Knuth

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