Class Note for MATH 796 at KU 7
Popular in Course
Popular in Department
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.
Reviews for Class Note for MATH 796 at KU 7
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
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
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'