Symmetric Functions, Alternating Sign Matrices, and Tokuyamas - PowerPoint PPT Presentation

Symmetric Functions, Alternating Sign Matrices, and Tokuyama’s Identity Angèle Hamel Wilfrid Laurier University Discrete Math Days/OCW May 23, 2015 Tuesday, 2 June, 15

Symmetric Functions Tuesday, 2 June, 15

Definition A function is symmetric if a permutation of its variables does not change the function. Tuesday, 2 June, 15

Examples x 2 1 + x 2 2 + x 2 3 + x 2 h 2 ( x 1 , x 2 , x 3 , x 4 ) = 4 + x 1 x 2 + x 1 x 3 + x 1 x 4 + x 2 x 3 + x 2 x 4 + x 3 x 4 x 3 1 + x 3 2 + x 3 3 + x 2 1 x 2 + x 2 1 x 3 + x 2 h 3 ( x 1 , x 2 , x 3 ) = 2 x 1 + x 2 2 x 3 + x 2 3 x 1 + x 2 3 x 2 + x 1 x 2 x 3 e 3 ( x 1 , x 2 , x 3 , x 4 ) = x 1 x 2 x 3 + x 1 x 2 x 4 + x 1 x 3 x 4 + x 2 x 3 x 4 Tuesday, 2 June, 15

Basis of Symmetric Functions Symmetric functions form a ring, and in fact, there is even more structure than that: you can find a basis. There are a number of great choices for a basis--elementary symmetric functions, e k , homogeneous symmetric functions, h k . But where’s the combinatorics?..... Tuesday, 2 June, 15

Partitions Given a partition, λ , with parts λ 1, λ 2 ,..., λ k , can be represented graphically by a diagram: + ≤ ≤ when λ = (4 , 3 , 3) = Tuesday, 2 June, 15

Tableaux Fill diagram with entries according to the following rules: entries weakly increase across rows entries strictly increase down columns 1 1 2 4 = 2 3 3 4 4 5 Tuesday, 2 June, 15

Weighting Tableaux Weight each entry i in the tableau by x i Then each tableau has weight x w 1 1 x w 2 2 · · · x w n n For example, the weight of 1 1 2 4 this tableau is = 2 3 3 x 2 1 x 2 2 x 2 3 x 3 4 x 5 4 4 5 Tuesday, 2 June, 15

Schur Functions � x wgt( T ) s λ ( x ) = T ∈ T λ ( n ) Tuesday, 2 June, 15

1 1 1 1 1 2 1 2 2 3 2 3 x 1 x 2 x 2 x 2 x 1 x 2 x 3 1 x 3 1 x 2 2 2 2 2 3 1 3 1 3 3 3 3 2 x 2 x 1 x 2 x 2 x 2 x 1 x 2 x 3 2 x 3 3 3 Tuesday, 2 June, 15

Schur function: s 21 (x 1 ,x 2 ,x 3 ) s 21 ( x 1 , x 2 , x 3 ) = x 2 1 x 2 + x 2 1 x 3 + x 1 x 2 2 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 3 + x 2 x 2 3 + x 2 2 x 3 Tuesday, 2 June, 15

Alternating Sign Matrices Tuesday, 2 June, 15

Alternating Sign Matrix Square matrices with entries from 0, 1, or -1 Each row and column contains at least one 1; first and last nonzero elements of each row and column are 1 Nonzero entries in each row and column alternate in sign 13 Tuesday, 2 June, 15

Alternating Sign Matrix Alternating sign matrices (ASM) generalize permutation matrices 14 Tuesday, 2 June, 15

Example   0 0 1   1 0 0 0 1 0 0 1 0     1 0 0 0 0 1     0 1 0 0 1 0   0 1 0 0 0 1 1 0 0 1 − 1 1       1 0 0 0 0 1 0 1 0     0 0 1 1 0 0 1 0 0 0 0 1     0 1 0 0 1 0 Tuesday, 2 June, 15

Alternating Sign Matrix × The number A( m ) of mxm ASM is: m − 1 (3 j + 1)! � A ( m ) = ( m + j )! j = 0 This was the Alternating Sign Matrix Conjecture See D.M. Bressoud, Proof and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge UP: 1999 Tuesday, 2 June, 15

Side Quest Vandermonde Identity Tuesday, 2 June, 15

Vandermonde Identity x n − 1 x n − 2 1 0 1 x 1 1 1 · · · x n − 1 x n − 2 1 x 2 B C · · · 2 2 Y B C ( x i − x j ) = det . . . . B . . . . C . . . . B C 1 ≤ i<j ≤ n · · · @ A x n − 1 x n − 2 1 x n · · · n n ( − 1) σ x n − 1 σ (1) x n − 2 X = σ (2) . . . x σ ( n − 1) σ ∈ S n LHS: product RHS: sum over of choices permutations Tuesday, 2 June, 15

Tournaments 2 2 2 2 2 1 1 3 1 3 1 3 1 3 2 2 2 2 3 1 1 3 1 3 1 3 Tuesday, 2 June, 15

Weighted Tournaments Weight each edge coming from node i by x i 2 x 2 1 x 2 1 3 Tuesday, 2 June, 15

Weighted Tournaments 2 2 2 2 2 x 2 x 2 x 2 x 1 x 2 x 3 2 x 3 3 x 1 3 x 2 1 1 3 1 3 3 1 3 1 2 2 2 2 x 2 x 1 x 2 x 3 x 2 x 2 1 x 2 1 x 3 2 x 1 3 1 1 3 1 3 1 3 Tuesday, 2 June, 15

Transitive Tournaments (bad!) 2 2 2 2 2 x 2 x 2 x 2 x 1 x 2 x 3 2 x 3 3 x 1 3 x 2 1 1 3 1 3 3 1 3 1 2 2 2 2 x 2 x 1 x 2 x 3 x 2 x 2 1 x 2 1 x 3 2 x 1 3 1 1 3 1 3 1 3 Tuesday, 2 June, 15

Tournaments and Permutations 2 2 2 x 2 x 2 x 2 3 x 1 3 x 2 2 x 3 1 3 1 3 3 1 2 2 2 x 2 x 2 x 2 1 x 2 1 x 3 2 x 1 3 1 1 3 1 3 Each x term ! 1 2 3 n · · · σ = σ (1) σ (2) σ (3) σ ( n ) corresponds to · · · = x n − 1 σ (1) x n − 2 a permutation: ⇒ σ (2) . . . x σ ( n − 1) Tuesday, 2 June, 15

Vandermonde Identity (Gessel, 1979) x n − 1 x n − 2 1 0 1 x 1 1 1 · · · x n − 1 x n − 2 1 x 2 B C · · · 2 2 Y B C ( x i − x j ) = det . . . . B . . . . C . . . . B C 1 ≤ i<j ≤ n · · · @ A x n − 1 x n − 2 1 x n · · · n n ( − 1) σ x n − 1 σ (1) x n − 2 X = σ (2) . . . x σ ( n − 1) σ ∈ S n LHS: product RHS: sum over of choices permutations LHS: direction of RHS: weight of edge ij in tournament tournament Tuesday, 2 June, 15

But what about the transitive tournaments? They have a weight too--it just doesn’t correspond to a permutation. But does it correspond to something else?.... Tuesday, 2 June, 15

Tournaments and ASM (Robbins and Rumsey, 1986) ( − 1) σ x n − 1 σ (1) x n − 2 Y X ( x i − x j ) = σ (2) . . . x σ ( n − 1) 1 ≤ i<j ≤ n σ ∈ S n n x NE i ( A )+ SE i ( A )+ NS i ( A ) t SE ( A ) (1 + t ) NS ( A ) Y X Y ( x i + tx j ) = i 1 ≤ i<j ≤ n i =1 A ∈ A n Tuesday, 2 June, 15

Tokuyama’s Identity Tuesday, 2 June, 15

Tokuyama’s Identity Proved by Tokuyama in 1988 using representation theory of general linear groups Proved by Okada in 1990 using algebraic manipulations on monotone triangles (equivalent to alternating sign matrices) Tuesday, 2 June, 15

Playing with Identities Tokuyama’s identity: n t hgt( ST ) (1 + t ) str( ST ) − n x wgt( ST ) � � � x i ( x i + tx j ) s λ ( x ) = i = 1 1 ≤ i < j ≤ n ST ∈ ST µ ( n ) t-deformation of a Weyl denominator formula Tuesday, 2 June, 15

Shifted Tableaux weakly increasing in rows weakly increasing down columns strictly increasing down left-to-right diagonals 1 1 1 2 2 2 3 3 5 2 2 3 3 4 5 5 6 3 3 4 4 5 6 ST = ∈ 4 5 5 5 5 6 6 6 Tuesday, 2 June, 15

Shifted Tableaux 1 1 1 2 2 2 3 3 5 2 2 3 3 4 5 5 6 wgt( ST ) = (3 , 5 , 6 , 4 , 8 , 5) 3 3 4 4 5 6 ∈ ST 986431 (6) with ST = str( ST ) = 12 , hgt( ST ) = 6 . 4 5 5 5 5 6 6 6 wgt(ST)= weight of the shifted tableau str(ST)= disjoint connected components of ribbon strips hgt(ST)= height of the tableau Tuesday, 2 June, 15

Back to ASM: μ -ASM μ = μ 1 , μ 2 , ..., μ k is a partition Rectangular matrices with entries from 0, 1, or -1 Nonzero entries in each row and column alternate in sign Each row and column contains at least one 1; first and last nonzero elements of each row are 1 First nonzero element in each column is 1 Last nonzero element is 1 in column q if q= μ i for some i, and 0 otherwise Tuesday, 2 June, 15

ASM statistics  0 0 1 0 0 0 0 0 0  0 0 0 1 0 0 0 0 0       1 0 − 1 0 0 1 0 0 0   A =   0 0 0 0 1 − 1 0 0 1       0 0 1 − 1 0 0 0 1 0   0 0 0 1 − 1 1 0 0 0 Four kinds of zeros: NE, SW, NW, SE Two kinds of ones: WE (+1s), NS (-1s) Tuesday, 2 June, 15

 0 0 1 0 0 0 0 0 0  0 0 0 1 0 0 0 0 0       1 0 − 1 0 0 1 0 0 0   A =   0 0 0 0 1 − 1 0 0 1       0 0 1 − 1 0 0 0 1 0   0 0 0 1 − 1 1 0 0 0   N E N E W E NW NW NW NW NW NW   N E N E SE W E NW NW NW NW NW     W E NW N S SE N E W E NW NW NW       SE N E N E SE W E N S N E N E W E     SE N E W E N S SE N E N E W E SW   SE N E SE W E N S W E NW SW SW Tuesday, 2 June, 15

Tokuyama for ASM H. and King, 2007 (generalization of Chapman, 2001): = n x N E k ( A ) y SE k ( A ) � � � ( x k + y k ) N S k ( A ) . ( x i + y j ) s λ ( x ) = k k A ∈ A µ ( n ) 1 ≤ i < j ≤ n k = 1 Or, if you like t’ s.... n t SE k ( A ) (1+ t ) NS k ( A ) x NE k ( A )+ SE k ( A )+ NS k ( A ) Y X Y ( x i + tx j ) s λ ( x ) = k 1 ≤ i<j ≤ n A ∈ A µ ( n ) k =1 Tuesday, 2 June, 15

Primed Shifted Tableaux weak increase across each row 1 2 � 2 1 1 2 3 3 5 2 3 � 3 4 � 5 � 5 6 � 2 weak increase down each column 3 4 � 4 5 � 6 3 = 4 5 � 5 no two identical unprimed entries in 5 5 6 � 6 any column 6 no two identical primed entries in any wgt( PST ) (3 row Tuesday, 2 June, 15