Pfaffian Formulae and Their Applications to Symmetric Function Identities. Soichi OKADA (Nagoya University)

Transcription

1 Pfaffan Formulae and Ther Applcatons to Symmetrc Functon Identtes Soch OKADA Nagoya Unversty 73rd Sémnare Lotharngen de Combnatore Strobl, September 9, 2014

2 Schur-type Pfaffans

3 Pfaffan Let A = a j 1, j 2m be a 2m 2m skew-symmetrc matrx. The Pfaffan of A s defned by Pf A = sgnπa π1,π2 a π3,π4 a π2m 1,π2m, π F 2m where F 2m s the subset of the symmetrc group S 2m gven by π1 < π3 < < π2m 1 F 2m = π S 2m :, π2 π4 π2m and sgnπ denotes the sgnature of π. For example, f 2m = 4, then Pf > > > 0 a 12 a 13 a 14 a 12 0 a 23 a 24 a 13 a 23 0 a 34 = a 12a 34 a 13 a 24 + a 14 a 23. a 14 a 24 a 34 0

4 Schur-type Pfaffans Schur 1911 xj x Pf x j + x 1, j n = 1 <j n x j x x j + x. Laksov Lascoux Thorup 1989, Stembrdge 1990 xj x Pf = x j x. 1 x x j 1 x x j 1, j n 1 <j n Knuth 1996 x j x Pf c + bx + x j + ax x j where b 2 = ac ± 1. 1, j n = 1 <j n x j x c + bx + x j + ax x j,

5 Sundqust 1996, Okada 1998 aj a b j b Pf x j x 1, j n 1 = 1 <j n x j x det Ṽ n/2,n/2 x; a det Ṽ n/2,n/2 x; b, where x = x 1,, x n, a = a 1,, a n, b = b 1,, b n and Ṽn p,q x; a = 1, x, x 2 }, {{, xp 1, a }, a x, a x 2,, a x q 1 }{{ } If we replace then we recover Schur s Pfaffan. p x by x 2, a by x, and b by x, q 1 n.

6 Theorem A If n + p + q = 2m s even and n p + q, then we have Sn x; a, b Ṽ p,q n x; b Pf t Ṽn p,q x; b O 1p q 2 +m pq m,m p q = 1 <j n x det Ṽn x; a det j x Ṽ n m q,m p x; b, where aj a b j b S n x; a, b =, x j x Ṽ p,q n x; a = 1, j n 1, x, x 2 }, {{, xp 1, a }, a x, a x 2,, a x q 1 }{{ } p The case p = q = 0 s the Sundqust Okada Pfaffan. q 1 n.

7 Theorem B If n + r = 2m s even and n r, then we have aj a 1, x Pf 1 x x, x 2,, xr 1 j 1, j n t 1, x, x 2,, xr 1 O = 1 n 1 n 1 m 2 + r 2 1 <j n 1 x x j det x m 1, x m + x m 2, x m+1 + x m 3,, x 2m 2 + 1, }{{} m a x m 1, a x m + x m 2,, a x n 2 + x r }{{ } m r 1 n. If r = 0 and a = x 1 n, then ths reduces to the LLTS Pfaffan.

8 Idea of the Proof Theorem B follows from Theorem A wth p = q or p = q + 1 by substtutng x by x + x 1, and b by x. Theorem A s proved by comparng the coeffcent of b I = I b on the both sdes, and the proof s reduced to showng c d j 1, z z det w, z 2,, zl 1 j 1 r, 1 j s 1 r t 1, w j, w 2 j,, wk 1 j = 1 j s 1 s+ r k 2 r=1 sj=1 z w j O s+l,r l det Ṽn z w; c d.

9 Applcatons to Symmetrc Functon Identtes 1. Theorem A Schur s P -functons. 2. Thoerem B restrcted Lttlewood s formula.

10 Applcatons to Schur s P -functons Schur s P -functons P λ x or Q-functons Q λ x are symmetrc functons, whch play a fundamental role n the theory of projectve representatons of the symmetrc groups, smlar to that of Schur functons s λ x n the theory of lnear representatons. Nmmo gave a formula for P λ x 1,, x n n terms of a Pfaffan. Let λ be a strct partton of length l,.e., λ 1 > λ 2 > > λ l > 0. If n + l s even, then we have P λ x = 1 <j n x + x j x x j Pf x x j x + x j 1, j n A smlar formula holds n the case where n + l s odd. x λ l, xλ l 1,, x λ 1 1 n O.

11 On the other hand, by replacng x by x 2, a by x, and b by x, the left hand sde of the Pfaffan formula n Theorem A reads xj x x Pf j + x 1, j n 1, x 2, x4,, x2p 1, x, x 3, x5,, x2q 1+1 O 1 n Comparng ths wth Nmmo s formula, we obtan an algebrac proof of Theorem Then we have In partcular, we have Worley; Conj. by Stanley We put ρ k = k, k 1,, 2, 1. P ρk +ρ l x = s ρk xs ρl x. P ρk x = s ρk x..

12 Smlarly, by specalzng x x 2, a x 1 + tx, b x n Theorem A, and equatng the coeffcents of t l, we can prove Theorem Worley We put ρ k = k, k 1,, 2, 1, and 1 l = 1, } {{, 1 }. l If 0 l k + 1, then we have P ρk +1 l x = s λ x, λ where λ runs over all parttons satsfyng ρ k λ ρ k+1 and λ ρ k = l.

13 Applcatons to restrcted Lttlewood s formulae Theorem Schur, Lttlewood 1 s λ x 1,, x n = n=1 1 x 1 <j n 1 x x j, λ where λ runs over all parttons. Theorem Kng; Conj. by Levens Stolova Van der Jeugt 1 s λ x 1,..., x n = n=1 1 x lλ l 1 <j n 1 x x j det x n j 1 l χ[j > l]x n l+j 1 det x n j 1, j n 1, j n where λ runs over all parttons of length l, and χ[j > l] = 1 f j > l and 0 otherwse.,

14 We can gve another proof to Kng s formula by usng the mnorsummaton formula Ishkawa Wakayama and Schur-type Pfaffan formula Theorem B. Theorem Ishkawa Wakayama Suppose that n + r s even and 0 n r N. For a matrx T = t j 1 n, 1 j r+n = [1, r] [r + 1, r + N] H K and a skew-symmetrc matrx A = a j r+1, j r+n, we have Pf AJ det H K[J] = 1 rr 1/2 KA Pf t K H t, H O J where J = {j 1 < < j n r } runs over all n r-element subsets of [r + 1, r + N] and AJ = a jp,j q 1 p, q n r, K[J] = t p,jq 1 p n, 1 q n r.

15 Outlne of the Proof of Kng s formula For smplcty, we assume that l s even. We put r = n l. Frst we apply the mnor-summaton formula to the matrces T = x 1 x 2 1 x x 2 x 2 2 x x n x 2 n x 3 n, A = wth H = x j 1 n, 0 j r 1 and K = x j express the summaton s λ x 1,..., x n n terms of a Pfaffan. lλ l r r + 1 r + 2 r n, j r. Then we can

16 Parttons of length n are n bjecton wth n-element subsets of nonnegatve ntegers, va λ I n λ = {λ 1 + n 1, λ 2 + n 2,, λ n 1 + 1, λ n }. Then we have lλ l [0, n l 1] I n λ. If lλ l and J = I n λ \ [0, n l 1], then det H K[J] s λ x 1,, x n = 1 <j n x, Pf AJ = 1. j x Hence, by applyng the mnor-summaton formula, we have 1 rn r KA s λ x 1,, x n = 1 <j n x j x Pf t K H t. H O lλ l

17 The, j entry of KA t K s gven by x r xr j x j x 1 x 1 x j 1 x x j = xr xr j 1 x x j xj x. 1 x j 1 x Hence we have 1 s λ x 1,, x n = rn r lλ l 1 <j n x j x x r x r j a j a 1, x 1 x Pf x, x 2,, xr 1 j 1, j n t 1, x, x 2,, xr 1 O 1 n 1 n where a = x /1 x. Now we can use Theorem B to convert the Pfaffan nto a determnant.,

18 Varatons By applyng the mnor-summaton formula to approprate matrces T and A, and usng Theorem B, we have Theorem λ E, lλ l t λ E, lλ 2l s λ x= s λ x= Kng 1 n =1 1 x2 1 <j n 1 x x j det x n j 1 1 <j n 1 x x j χ[j l]x n l+j det + det x n j + χ[j l]x n l+j, 2 det x n j + χ[j > 2l + 1]x n 2l+j 2, det x n j x n j where E s the set of even parttons,.e., parttons wth only even parts.