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

• f its variables does not change the

function.

Tuesday, 2 June, 15

Examples

h2(x1, x2, x3, x4) = x2

1 + x2 2 + x2 3 + x2 4 + x1x2 + x1x3

+x1x4 + x2x3 + x2x4 + x3x4 h3(x1, x2, x3) = x3

1 + x3 2 + x3 3 + x2 1x2 + x2 1x3 + x2 2x1

+x2

2x3 + x2 3x1 + x2 3x2 + x1x2x3

e3(x1, x2, x3, x4) = x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4

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, ek, homogeneous symmetric functions, hk. But where’s the combinatorics?.....

Tuesday, 2 June, 15

Partitions

Given a partition, λ, with parts λ1,λ2,...,λk, 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 Then each tableau has weight For example, the weight of this tableau is

x2

1x2 2x2 3x3 4x5

=

1 1 2 4 2 3 3 4 4 5

xw1

1 xw2 2 · · · xwn n

xi

Tuesday, 2 June, 15

Schur Functions

sλ(x) =

• T ∈T λ(n)

xwgt(T )

Tuesday, 2 June, 15

1 1 2

1 1 3 1 2 2 1 2 3

1 3 2

1 3 3

2 2 3

2 3 3

x2

1x2

x2

1x3

x1x2

2

x1x2

3

x1x2x3

x1x2x3

x2

2x3

x2x2

3

Tuesday, 2 June, 15

Schur function: s21(x1,x2,x3)

s21(x1, x2, x3) = x2

1x2 + x2 1x3 + x1x2 2 + x1x2x3

+x1x2x3 + x1x2

3 + x2x2 3 + x2 2x3

Tuesday, 2 June, 15

Alternating Sign Matrices

Tuesday, 2 June, 15

Alternating Sign Matrix

Square matrices with entries from 0, 1,

• r -1

Each row and column contains at least

• ne 1; first and last nonzero elements
• f each row and column are 1

Nonzero entries in each row and column alternate in sign

13

Tuesday, 2 June, 15

Alternating Sign Matrix

14

Alternating sign matrices (ASM) generalize permutation matrices

Tuesday, 2 June, 15

Example

  1 1 1     1 1 1     1 1 1     1 1 1     1 1 1     1 1 1     1 1 −1 1 1  

Tuesday, 2 June, 15

Alternating Sign Matrix

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

The number A(m) of mxm ASM is:

× A(m) =

m−1

• j=0

(3 j + 1)! (m + j)!

Tuesday, 2 June, 15

Side Quest

Vandermonde Identity

Tuesday, 2 June, 15

Vandermonde Identity

Y

1≤i<j≤n

(xi − xj) = det

B B B B @

xn−1

1

xn−2

1

· · · x1 1 xn−1

2

xn−2

2

· · · x2 1 . . . . . . . . . · · · . . . xn−1

n

xn−2

n

· · · xn 1

1 C C C C A

=

X

σ∈Sn

(−1)σxn−1

σ(1)xn−2 σ(2) . . . xσ(n−1)

LHS: product

• f choices

RHS: sum over permutations

Tuesday, 2 June, 15

Tournaments

1 1 1 1 1 1 1 1 2 1 2 2 2 2 2 2 2 3 2 3 3 3 3 3 3 3

Tuesday, 2 June, 15

Weighted Tournaments

1 2 3

x2

1x2

Weight each edge coming from node i by xi

Tuesday, 2 June, 15

Weighted Tournaments

1 1 1 1 1 1 1 2 1 2 2 2 2 2 2 3 2 3 3 3 3 3 3 1 2 3

x2

1x2

x2

1x3

x2

2x1

x2

2x3

x2

3x1

x2

3x2

x1x2x3

x1x2x3

Tuesday, 2 June, 15

Transitive Tournaments (bad!)

1 1 1 1 1 1 1 2 1 2 2 2 2 2 2 3 2 3 3 3 3 3 3 1 2 3

x2

1x2

x2

1x3

x2

2x1

x2

2x3

x2

3x1

x2

3x2

x1x2x3

x1x2x3

Tuesday, 2 June, 15

Tournaments and Permutations

1 1 2 2 3 3 1 2 3

x2

1x2

x2

1x3

x2

2x1

1 1 1 2 2 3 2 3 3

x2

2x3

x2

3x1

x2

3x2

Each x term corresponds to a permutation:

σ = 1 2 3 · · · n σ(1) σ(2) σ(3) · · · σ(n)

!

= ⇒

xn−1

σ(1)xn−2 σ(2) . . . xσ(n−1)

Tuesday, 2 June, 15

Vandermonde Identity (Gessel, 1979)

Y

1≤i<j≤n

(xi − xj) = det

B B B B @

xn−1

1

xn−2

1

· · · x1 1 xn−1

2

xn−2

2

· · · x2 1 . . . . . . . . . · · · . . . xn−1

n

xn−2

n

· · · xn 1

1 C C C C A

=

X

σ∈Sn

(−1)σxn−1

σ(1)xn−2 σ(2) . . . xσ(n−1)

LHS: product

• f choices

RHS: sum over permutations LHS: direction of edge ij in tournament RHS: weight of 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)

Y

1≤i<j≤n

(xi − xj) =

X

σ∈Sn

(−1)σxn−1

σ(1)xn−2 σ(2) . . . xσ(n−1)

Y

1≤i<j≤n

(xi + txj) =

X

A∈An

tSE(A)(1 + t)NS(A)

n

Y

i=1

xNEi(A)+SEi(A)+NSi(A)

i

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

• i=1

xi

• 1≤i< j≤n

(xi + tx j) sλ(x) =

• ST ∈ST µ(n)

thgt(ST )(1 + t)str(ST )−n xwgt(ST )

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

ST =

1 1 1 2 2 2 3 3 5 2 2 3 3 4 5 5 6 3 3 4 4 5 6 4 5 5 5 5 6 6 6

∈

Tuesday, 2 June, 15

Shifted Tableaux

wgt(ST)=weight of the shifted tableau str(ST)=disjoint connected components

• f ribbon strips

hgt(ST)=height of the tableau

ST =

1 1 1 2 2 2 3 3 5 2 2 3 3 4 5 5 6 3 3 4 4 5 6 4 5 5 5 5 6 6 6

∈ ST 986431(6) with wgt(ST)=(3, 5, 6, 4, 8, 5) str(ST)=12, hgt(ST )=6.

Tuesday, 2 June, 15

Back to ASM: μ-ASM

μ=μ1, μ2, ..., μk 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=μi for some i, and 0 otherwise

Tuesday, 2 June, 15

ASM statistics

Four kinds of zeros: NE, SW, NW, SE Two kinds of ones: WE (+1s), NS (-1s)

A =           1 1 1 −1 1 1 −1 1 1 −1 1 1 −1 1          

Tuesday, 2 June, 15

A =           1 1 1 −1 1 1 −1 1 1 −1 1 1 −1 1          

         

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

• H. and King, 2007 (generalization of

Chapman, 2001):

Tokuyama for ASM

Or, if you like t’s....

Y

1≤i<j≤n

(xi+txj) sλ(x) = X

A∈Aµ(n) n

Y

k=1

tSEk(A)(1+t)NSk(A)xNEk(A)+SEk(A)+NSk(A)

k

=

• 1≤i< j≤n

(xi + y j) sλ(x) =

• A∈Aµ(n)

n

• k=1

x N Ek(A)

k

ySEk(A)

k

(xk + yk)N Sk(A).

Tuesday, 2 June, 15

Primed Shifted Tableaux

weak increase across each row weak increase down each column no two identical unprimed entries in any column no two identical primed entries in any row

=

1 1 1 2 2 2 3 3 5 2 2 3 3 4 5 5 6 3 3 4 4 5 6 4 5 5 5 5 6 6 6

wgt(PST) (3

Tuesday, 2 June, 15

Proof idea...

Use an association between ASM and primed shifted tableaux...

PST =

1 1 1 2 3 3 4 4 4 2 2 2 3 4 5 5 5 3 4 4 4 5 6 4 5 5 6 5 6 6 6

= ⇒ M(PST )=          1 1 1 0 0 0 0 2 2 2 2 0 0 0 0 3 0 0 3 3 3 0 0 0 4 4 4 4 4 0 4 4 4 5 5 5 0 5 5 5 5 0 6 6 6 6 0 6 0 0 0         

Tuesday, 2 June, 15

PST =

1 1 1 2 3 3 4 4 4 2 2 2 3 4 5 5 5 3 4 4 4 5 6 4 5 5 6 5 6 6 6

= ⇒ M(PST )=          1 1 1 0 0 0 0 2 2 2 2 0 0 0 0 3 0 0 3 3 3 0 0 0 4 4 4 4 4 0 4 4 4 5 5 5 0 5 5 5 5 0 6 6 6 6 0 6 0 0 0          A =           1 1 1 −1 1 1 −1 1 1 −1 1 1 −1 1          

= ⇒

...and use jeu de taquin on the primed shifted tableau...

Tuesday, 2 June, 15

...to create a pair of tableaux

1 2 1 4 5 6 1 2 3 2 3 2 5 2 3 5 5 3 4 3 3 4 6 4 5 6 5 5 5 6 6

≡

1 2 1 4 5 6 2 3 2 5 2 3 4 3 3 4 5 6 5 5 6

·

1 2 3 3 5 5 4 6 5 6

One corresponding to ...and the other corresponding to

• 1≤i≤ j≤n

(xi + y j)

= sλ(x)

Tuesday, 2 June, 15

Other Stuff

Six vertex model

Tuesday, 2 June, 15

Square Ice

So-called because it models in a two dimensional grid the orientation of molecules in frozen water. Also called the six-vertex model.

↑ ↓ ↑ ↓ ↑ ↓ → · ← ← · → → · → ← · ← ← · ← → · → ↓ ↑ ↑ ↓ ↑ ↓

Tuesday, 2 June, 15

Tuesday, 2 June, 15

A =           1 1 1 −1 1 1 −1 1 1 −1 1 1 −1 1          

Tuesday, 2 June, 15

Other Stuff

Symplectic and Orthogonal Characters

Tuesday, 2 June, 15

Half-turn ASM

              1 1 1 1 1 1 1 1 1 1                            

SE SE SE SE WE SE SE SE NS SW WE SE WE SE NE SE NE SE NE SE NS SW NW WE WE SE NE NE NS WE NE NE WE NE NE NE

             

• 1

2 3 4 4 3 2 1

• Tuesday, 2 June, 15

X X X X X X X X

Six Conjectures

like a Schur fn Sum over half-turn alternating sign matrices like a Vandermonde

X

A∈Aδ

Xn

wgt(A) ∆µ

Xn(x; t) =

X

A∈Aλ

Xn

wgt(A)

where X

A∈Aδ

Xn

wgt(A) = ΦXn(x; t) and = +µ

Symplectic and Orthogonal Characters

Tuesday, 2 June, 15

Other Stuff

Factorial Schur functions

Tuesday, 2 June, 15

Factorial Schur Functions

sum is over all tableaux of shape λ, and c(α) is the content of the square (c(α)=j- i for square α).

sλ(x|a) =

• T
• α∈λ

(xT(α) − aT(α)+c(α))

Tuesday, 2 June, 15

Weighted Tableaux

Weight each entry k in position i, j by =

1 1 2 4 2 3 3 4 4 5

(x1 − a1) (x1 − a2) (x2 − a4) (x4 − a7) (x2 − a1) (x3 − a3) (x3 − a4) (x4 − a2) (x4 − a3) (x5 − a5)

xk − ak+j−i

Tuesday, 2 June, 15

Factorial Tokuyama (Bump et al., 2010; new proof H. and King, 2015)

Pλ(x; y|a) = Y

16i6n

xi Y

16i<j6n

(xi + yj) sµ(x|a) ; Y

Tuesday, 2 June, 15

Bibliography

• D. Bump, P.J. McNamara, M. Nakasuji,

Factorial Schur functions and the Yang- Baxter equation, arXiv: 1108.3087, 2011.

• R. Chapman, Alternating sign matrices and

tournaments, Adv. Appl. Math. 27 (2001), 318-335. I.M. Gessel, Tournaments and Vandermonde’s Determinant, J. Graph Theory, 3 (1979), 305-307.

Tuesday, 2 June, 15

A.M. Hamel, R.C. King, Bijective proofs of shifted tableau and alternating sign matrix identities, J. Alg. Comb. 25 (2007), 417-458.

A.M. Hamel and R.C. King, Tokuyama’s identity for factorial Schur P and Q functions, Electronic J. Combinatorics, under review, 2015. A.M. Hamel and R.C. King, Half-turn symmetric alternating sign matrices and Tokuyama type factorisation for orthogonal group characters, JCTA, 131 (2015), 1-31.

Tuesday, 2 June, 15

• S. Okada, Partially strict shifted plane

partitions, JCTA, 53 (1990), 143-156.

• S. Okada, Alternating sign matrices and some

deformations of Weyl’s denominator formula,

• J. Alg. Comb., 2 (1993), 155-176.

D.P. Robbins, H. Rumsey, Determinants and alternating sign matrices, Adv. Math. 62 (1986), 169-184.

• T. Tokuyama, A generating function of strict

Gelfand patterns and some formulas on characters of general linear group, J. Math.

• Soc. Japan, 40 (1988), 671-685.

Tuesday, 2 June, 15