Exploiting Symmetries

David M. BressoudMacalester College, St. Paul, MNTalk given at University of FloridaOctober 29, 2004

1. The Vandermonde determinant

2. Weyl’s character formulae

3. Alternating sign matrices

4. The six-vertex model of statistical mechanics

5. Okada’s work connecting ASM’s and character formulae

x1n−1 x2

n−1 L xnn−1

M M O M

x1 x2 L xn

1 1 L 1

= −1( )I σ( ) xin−σ i( )

i=1

n

∏σ∈Sn

∑

Cauchy 1815

“Memoir on functions whose values are equal but of opposite sign when two of their variables are interchanged”

(alternating functions) Augustin-Louis

Cauchy (1789–1857)

Cauchy 1815

“Memoir on functions whose values are equal but of opposite sign when two of their variables are interchanged”

(alternating functions)

This function is 0 when so it is divisible by

xi =xjxi −xj( )

i< j∏

x1n−1 x2

n−1 L xnn−1

M M O M

x1 x2 L xn

1 1 L 1

= −1( )I σ( ) xin−σ i( )

i=1

n

∏σ∈Sn

∑

Cauchy 1815

“Memoir on functions whose values are equal but of opposite sign when two of their variables are interchanged”

(alternating functions)

This function is 0 when so it is divisible by

xi =xjxi −xj( )

i< j∏

But both polynomials have same degree, so ratio is constant, = 1.

= xi − x j( )i< j∏

x1n−1 x2

n−1 L xnn−1

M M O M

x1 x2 L xn

1 1 L 1

= −1( )I σ( ) xin−σ i( )

i=1

n

∏σ∈Sn

∑

Cauchy 1815

Any alternating function in divided by the Vandermonde determinant yields a symmetric function:

x1, x2 ,K , xn

x1λ1 +n−1 x2

λ1 +n−1 L xnλ1 +n−1

M M O M

x1λn−1 +1 x2

λn−1 +1 L xnλn−1 +1

x1λn x2

λn L xnλn

x1n−1 x2

n−1 L xnn−1

M M O M

x11 x2

1 L xn1

x10 x2

0 L xn0

=sλ x1,x2 ,K ,xn( )

Cauchy 1815

Any alternating function in divided by the Vandermonde determinant yields a symmetric function:

x1, x2 ,K , xn

Called the Schur function. I.J. Schur (1917) recognized it as the character of the irreducible representation of GLn indexed by λ.

x1λ1 +n−1 x2

λ1 +n−1 L xnλ1 +n−1

M M O M

x1λn−1 +1 x2

λn−1 +1 L xnλn−1 +1

x1λn x2

λn L xnλn

x1n−1 x2

n−1 L xnn−1

M M O M

x11 x2

1 L xn1

x10 x2

0 L xn0

=sλ x1,x2 ,K ,xn( )

Issai Schur (1875–1941)

sλ 1,1,K ,1( ) is the dimension of the representation

sλ 1,1,K ,1( ) =ρ + λ( )⋅rρ ⋅rr∈An−1

+∏

where ρ =n−12

,n−32

,K ,1−n2

⎛⎝⎜

⎞⎠⎟=12

rr∈An−1

+∑ ,

λ = λ1,λ2 ,K ,λn( ),

An−1+ = ei −ej 1≤i < j ≤n{ } ,

ei is the unit vector with 1 in the ith coordinate

Note that the symmetric group on n letters is the group of transformations of

An−1 = ei −ej 1≤i ≠ j ≤n{ }

Weyl 1939 The Classical Groups: their invariants and representations

Sp2n λ; rx( ) =

x1λ1 +n −x1

−λ1−n L xnλ1 +n −xn

−λ1−n

M O M

x1λn+1 −x1

−λn−1 L xnλn+1 −xn

−λn−1

x1n −x1

−n L xnn −xn

−n

M O M

x11 −x1

−1 L xn1 −xn

−1

Sp2n λ; rx( ) is the character of the irreducible representation, indexed by the partition λ, of the symplectic group (the subgoup of GL2n of isometries).

Hermann Weyl (1885–1955)

Sp2n λ;1r

( ) =ρ + λ( )⋅rρ ⋅rr∈Cn

+∏

where ρ = n,n−1,K ,1( ) =12

rr∈Cn

+∑ ,

λ = λ1,λ2 ,K ,λn( ),

Cn+ = ei ±ej 1≤i < j ≤n{ } U 2ei 1≤i ≤n{ } ,

ei is the unit vector with 1 in the ith coordinate

The dimension of the representation is

Weyl 1939 The Classical Groups: their invariants and representations

x1n −x1

−n L xnn −xn

−n

M O M

x11 −x1

−1 L xn1 −xn

−1

x1x2L xn( )n

xi −xj( )i< j∏

is a symmetric polynomial. As a polynomial in x1 it has degree n + 1 and roots at

±1, x j−1 for 2 ≤ j ≤ n

Weyl 1939 The Classical Groups: their invariants and representations

x1n −x1

−n L xnn −xn

−n

M O M

x11 −x1

−1 L xn1 −xn

−1

x1x2L xn( )n

xi −xj( )i< j∏

= xi2 −1( )

i∏ xixj −1( )

i< j∏

is a symmetric polynomial. As a polynomial in x1 it has degree n + 1 and roots at

±1, x j−1 for 2 ≤ j ≤ n

Weyl 1939 The Classical Groups: their invariants and representations: The Denominator Formulas

x1

n−12 −x1−n+ 12 L xn

n−12 −xn−n+ 12

M O M

x112 −x1

−12 L xn

12 −xn

−12

x1x2L xn( )n−12

xi −xj( )i< j∏

= xi −1( )i∏ xixj −1( )

i< j∏

x1n−1 + x1

−n+1 L xnn−1 + xn

−n+1

M O M

x10 + x1

−0 L xn0 + xn

−0

x1x2L xn( )n−1

xi −xj( )i< j∏

=2 xixj −1( )i< j∏

Desnanot-Jacobi adjoint matrix thereom (Desnanot for n ≤ 6 in 1819, Jacobi for general case in 1833M j

i is matrix M with row i and column j removed.

detM =detM1

1 ⋅detMnn −detMn

1 ⋅detM1n

detM1,n1,n

Given that the determinant of the empty matrix is 1 and the determinant of a 11 is the entry in that matrix, this uniquely defines the determinant for all square matrices.

Carl Jacobi (1804–1851)

detM =detM1

1 ⋅detMnn −detMn

1 ⋅detM1n

detM1,n1,n

detλ M =detM1

1 ⋅detMnn + λdetMn

1 ⋅detM1n

detM1,n1,n

det−1 M =detM( )

detλ aji−1( )

i, j=1

n= ai + λaj( )

1≤i< j≤n∏

David Robbins (1942–2003)

detM =detM1

1 ⋅detMnn −detMn

1 ⋅detM1n

detM1,n1,n

detλ M =detM1

1 ⋅detMnn + λdetMn

1 ⋅detM1n

detM1,n1,n

detλa bc d

⎛⎝⎜

⎞⎠⎟=ad+ λbc

detλ

a b cd e fg h j

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=aej + λ bdj + afh( ) + λ2 bfg+ cdh( ) + λ 3ceg

+ λ 1+ λ( )bde−1 fh

detλ

a1,1 a1,2 a1,3 a1,4

a2,1 a2,2 a2,3 a2,4

a3,1 a3,2 a3,3 a3,4

a4,1 a4,2 a4,3 a4,4

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=a1,1a2,2a3,3a4,4

+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )

+L sums over other permutations×λ inversion number

+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L

+ λ 3 1+ λ−1( )2a1,2a2,1a2,2

−1 a2,3a3,2a3,3−1a3,4a4,3 +L

detλ

a1,1 a1,2 a1,3 a1,4

a2,1 a2,2 a2,3 a2,4

a3,1 a3,2 a3,3 a3,4

a4,1 a4,2 a4,3 a4,4

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=a1,1a2,2a3,3a4,4

+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )

+L sums over other permutations×λ inversion number

+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L

+ λ 3 1+ λ−1( )2a1,2a2,1a2,2

−1 a2,3a3,2a3,3−1a3,4a4,3 +L

0 1 0 0

1 −1 1 00 1 −1 10 0 1 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

0 1 0 0

1 −1 1 00 0 0 10 1 0 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

detλ

a1,1 a1,2 a1,3 a1,4

a2,1 a2,2 a2,3 a2,4

a3,1 a3,2 a3,3 a3,4

a4,1 a4,2 a4,3 a4,4

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=a1,1a2,2a3,3a4,4

+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )

+L sums over other permutations×λ inversion number

+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L

+ λ 3 1+ λ−1( )2a1,2a2,1a2,2

−1 a2,3a3,2a3,3−1a3,4a4,3 +L

0 1 0 0

1 −1 1 00 1 −1 10 0 1 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

0 1 0 0

1 −1 1 00 0 0 10 1 0 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

detλ

a1,1 a1,2 a1,3 a1,4

a2,1 a2,2 a2,3 a2,4

a3,1 a3,2 a3,3 a3,4

a4,1 a4,2 a4,3 a4,4

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=a1,1a2,2a3,3a4,4

+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )

+L sums over other permutations×λ inversion number

+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L

+ λ 3 1+ λ−1( )2a1,2a2,1a2,2

−1 a2,3a3,2a3,3−1a3,4a4,3 +L

detλ xi, j( ) = λ Inv A( )

A= ai , j( )

∑ 1+ λ−1( )N A( )

xi, jai , j

i, j∏

Sum is over all alternating sign matrices, N(A) = # of –1’s

detλ

a1,1 a1,2 a1,3 a1,4

a2,1 a2,2 a2,3 a2,4

a3,1 a3,2 a3,3 a3,4

a4,1 a4,2 a4,3 a4,4

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=a1,1a2,2a3,3a4,4

+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )

+L sums over other permutations×λ inversion number

+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L

+ λ 3 1+ λ−1( )2a1,2a2,1a2,2

−1 a2,3a3,2a3,3−1a3,4a4,3 +L

detλ xi, j( ) = λ Inv A( )

A= ai , j( )

∑ 1+ λ−1( )N A( )

xi, jai , j

i, j∏

xi + λxj( ) = λ Inv A( ) 1+ λ−1( )

N A( )xj

n−i( )ai , j

i, j∏

A∈An

∑1≤i< j≤n∏

n

1

2

3

4

5

6

7

8

9

An

1

2

7

42

429

7436

218348

10850216

911835460

= 2 3 7

= 3 11 13

= 22 11 132

= 22 132 17 19

= 23 13 172 192

= 22 5 172 193 23

How many n n alternating sign matrices?

n

1

2

3

4

5

6

7

8

9

An

1

2

7

42

429

7436

218348

10850216

911835460

= 2 3 7

= 3 11 13

= 22 11 132

= 22 132 17 19

= 23 13 172 192

= 22 5 172 193 23

n

1

2

3

4

5

6

7

8

9

An

1

2

7

42

429

7436

218348

10850216

911835460

There is exactly one 1 in the first

row

n

1

2

3

4

5

6

7

8

9

An

1

1+1

2+3+2

7+14+14+7

42+105+…

There is exactly one 1 in the first

row

1

1 1

2 3 2

7 14 14 7

42 105 135 105 42

429 1287 2002 2002 1287 429

1

1 1

2 3 2

7 14 14 7

42 105 135 105 42

429 1287 2002 2002 1287 429

+ + +

1

1 1

2 3 2

7 14 14 7

42 105 135 105 42

429 1287 2002 2002 1287 429

+ + +

1 0 0 0 0

0 ? ? ? ?

0 ? ? ? ?

0 ? ? ? ?

0 ? ? ? ?

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

1

1 2/2 1

2 2/3 3 3/2 2

7 2/4 14 14 4/2 7

42 2/5 105 135 105 5/2 42

429 2/6 1287 2002 2002 1287 6/2 429

1

1 2/2 1

2 2/3 3 3/2 2

7 2/4 14 5/5 14 4/2 7

42 2/5 105 7/9 135 9/7 105 5/2 42

429 2/6 1287 9/14 2002 16/16 2002 14/9 1287 6/2 429

2/2

2/3 3/2

2/4 5/5 4/2

2/5 7/9 9/7 5/2

2/6 9/14 16/16 14/9 6/2

2

2 3

2 5 4

2 7 9 5

2 9 16 14 6

1+1

1+1 1+2

1+1 2+3 1+3

1+1 3+4 3+6 1+4

1+1 4+5 6+10 4+10 1+5

Numerators:

1+1

1+1 1+2

1+1 2+3 1+3

1+1 3+4 3+6 1+4

1+1 4+5 6+10 4+10 1+5

Conjecture 1:

Numerators:

An,k

An,k+1

=

n−2k−1

⎛⎝⎜

⎞⎠⎟+

n−1k−1

⎛⎝⎜

⎞⎠⎟

n−2n−k−1

⎛⎝⎜

⎞⎠⎟+

n−1n−k−1

⎛⎝⎜

⎞⎠⎟

Conjecture 1:

Conjecture 2 (corollary of Conjecture 1):

An,k

An,k+1

=

n−2k−1

⎛⎝⎜

⎞⎠⎟+

n−1k−1

⎛⎝⎜

⎞⎠⎟

n−2n−k−1

⎛⎝⎜

⎞⎠⎟+

n−1n−k−1

⎛⎝⎜

⎞⎠⎟

An =

3 j +1( )!n+ j( )!j=0

n−1

∏ =1!⋅4!⋅7!L 3n−2( )!

n!⋅n+1( )!L 2n−1( )!

Conjecture 2 (corollary of Conjecture 1):

An =

3 j +1( )!n+ j( )!j=0

n−1

∏ =1!⋅4!⋅7!L 3n−2( )!

n!⋅n+1( )!L 2n−1( )!

Exactly the formula found by George Andrews for counting descending plane partitions.

George Andrews Penn State

Conjecture 2 (corollary of Conjecture 1):

An =

3 j +1( )!n+ j( )!j=0

n−1

∏ =1!⋅4!⋅7!L 3n−2( )!

n!⋅n+1( )!L 2n−1( )!

Exactly the formula found by George Andrews for counting descending plane partitions. In succeeding years, the connection would lead to many important results on plane partitions.

George Andrews Penn State

A n; x( ) = xN A( )

A∈An

∑A 1;x( ) =1,

A 2;x( ) =2,

A 3;x( ) =6 + x,

A 4;x( ) =24 +16x+ 2x2 ,

A 5;x( ) =120 + 200x+ 94x2 +14x3 + x4 ,

A 6;x( ) =720 + 2400x+ 2684x2 +1284x3 + 310x4 + 36x5 + 2x6

A 7;x( ) =5040 + 24900x+ 63308x2 + 66158x3 + 38390x4 +13037x5

+ 2660x6 + 328x7 + 26x8 + x9

A n; x( ) = xN A( )

A∈An

∑A 1;x( ) =1,

A 2;x( ) =2,

A 3;x( ) =6 + x,

A 4;x( ) =24 +16x+ 2x2 ,

A 5;x( ) =120 + 200x+ 94x2 +14x3 + x4 ,

A 6;x( ) =720 + 2400x+ 2684x2 +1284x3 + 310x4 + 36x5 + 2x6

A 7;x( ) =5040 + 24900x+ 63308x2 + 66158x3 + 38390x4 +13037x5

+ 2660x6 + 328x7 + 26x8 + x9

xi + λxj( ) = λ Inv A( ) 1+ λ−1( )

N A( )xj

n−i( )ai , j

i, j∏

A∈An

∑1≤i< j≤n∏

A n;0( ) =n!

A n;1( ) =An =3i +1( )!n+ i( )!i=0

n−1

∏⎛

⎝⎜⎞

⎠⎟

A n;2( ) =2n(n−1)/2

A n; x( ) = xN A( )

A∈An

∑A 1;x( ) =1,

A 2;x( ) =2,

A 3;x( ) =6 + x,

A 4;x( ) =24 +16x+ 2x2 ,

A 5;x( ) =120 + 200x+ 94x2 +14x3 + x4 ,

A 6;x( ) =720 + 2400x+ 2684x2 +1284x3 + 310x4 + 36x5 + 2x6

A 7;x( ) =5040 + 24900x+ 63308x2 + 66158x3 + 38390x4 +13037x5

+ 2660x6 + 328x7 + 26x8 + x9

A n; 3( ) =3n n−1( )

2n n−1( )

3 j −i( ) +13 j −i( )1≤i, j≤n

j−i odd

∏ Conjecture:

(MRR, 1983)

A n;0( ) =n!

A n;1( ) =An =3i +1( )!n+ i( )!i=0

n−1

∏⎛

⎝⎜⎞

⎠⎟

A n;2( ) =2n(n−1)/2

Mills & Robbins (suggested by Richard Stanley) (1991)

Symmetries of ASM’s

A n( ) =3 j +1( )!n+ j( )!j=0

n−1

∏

AV 2n+1( ) = −3( )n2 3 j −i( ) +1

j −i + 2n+11≤i, j≤2n+12 j

∏⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

A n( )

AHT 2n( ) = −3( )n n−1( )/2 3 j −i( ) + 2j −i +ni, j

∏⎛

⎝⎜⎞

⎠⎟A n( )

AQT 4n( ) =AHT 2n( )⋅A n( )2

Vertically symmetric ASM’s

Half-turn symmetric ASM’s

Quarter-turn symmetric ASM’s

December, 1992

Zeilberger announces a proof that # of ASM’s equals

3 j +1( )!n+ j( )!j=0

n−1

∏

Doron Zeilberger

Rutgers University

December, 1992

Zeilberger announces a proof that # of ASM’s equals

3 j +1( )!n+ j( )!j=0

n−1

∏

1995 all gaps removed, published as “Proof of the Alternating Sign Matrix Conjecture,” Elect. J. of Combinatorics, 1996.

Zeilberger’s proof is an 84-page tour de force, but it still left open the original conjecture:

An,k

An,k+1

=

n−2k−1

⎛⎝⎜

⎞⎠⎟+

n−1k−1

⎛⎝⎜

⎞⎠⎟

n−2n−k−1

⎛⎝⎜

⎞⎠⎟+

n−1n−k−1

⎛⎝⎜

⎞⎠⎟

1996 Kuperberg announces a simple proof

“Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices

Greg Kuperberg

UC Davis

“Another proof of the alternating sign matrix conjecture,” International Mathematics Research NoticesPhysicists have been

studying ASM’s for decades, only they call them square ice (aka the six-vertex model ).

1996 Kuperberg announces a simple proof

H O H O H O H O H O H

H H H H H

H O H O H O H O H O H

H H H H H

H O H O H O H O H O H

H H H H H

H O H O H O H O H O H

H H H H H

H O H O H O H O H O H

Horizontal 1

Vertical –1

southwest

northeast

northwest

southeast

N = # of verticalI = inversion number = N + # of SW

x2, y3

Anatoli Izergin

Vladimir Korepin

SUNY Stony Brook

1980’s

det1

xi −yj( ) axi −yj( )

⎛

⎝⎜

⎞

⎠⎟

xi −yj( ) axi −yj( )i, j=1

n∏xi −xj( ) yi −yj( )1≤i< j≤n∏

= 1−a( )2N A( ) an(n−1)/2−Inv A( )

A∈An

∑

× xivert∏ yj axi −yj( )

SW, NE∏ xi −yj( )

NW, SE∏

Proof:LHS is symmetric polynomial in x’s and in y’s

Degree n – 1 in x1

By induction, LHS = RHS when x1 = y1

Sufficient to show that RHS is symmetric polynomial in x’s and in y’s

LHS is symmetric polynomial in x’s and in y’s

Degree n – 1 in x1

By induction, LHS = RHS when x1 = –y1

Sufficient to show that RHS is symmetric polynomial in x’s and in y’s — follows from Baxter’s triangle-to-triangle relation

Proof:

Rodney J. Baxter

Australian National University

a =z−4 , xi =z2 , yi =1

RHS= z−z−1( )n n−1( )

z+ z−1( )2N A( )

A∈An

∑

det1

xi −yj( ) axi −yj( )

⎛

⎝⎜

⎞

⎠⎟

xi −yj( ) axi −yj( )i, j=1

n∏xi −xj( ) yi −yj( )1≤i< j≤n∏

= 1−a( )2N A( ) an(n−1)/2−Inv A( )

A∈An

∑

× xivert∏ yj axi −yj( )

SW, NE∏ xi −yj( )

NW, SE∏

det1

xi −yj( ) axi −yj( )

⎛

⎝⎜

⎞

⎠⎟

xi −yj( ) axi + yj( )i, j=1

n∏xi −xj( ) yi −yj( )1≤i< j≤n∏

= 1−a( )2N A( ) an(n−1)/2−Inv A( )

A∈An

∑

× xivert∏ yj axi −yj( )

SW, NE∏ xi −yj( )

NW, SE∏

z =eπ i /3 : RHS= −3( )n n−1( )/2 An ,

z=eπ i /4 : RHS= −2( )n n−1( )/2 2N A( )

A∈An

∑ ,

z=eπ i /6 : RHS= −1( )n n−1( )/2 3N A( )

A∈An

∑ .

a =z−4 , xi =z2 , yi =1

RHS= z−z−1( )n n−1( )

z+ z−1( )2N A( )

A∈An

∑

1996

Doron Zeilberger uses this determinant to prove the original conjecture

“Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics

2001, Kuperberg uses the power of the triangle-to-triangle relation to prove some of the conjectured formulas:

AV 2n +1( ) = −3( )n2 3 j −i( ) +1

j −i + 2n+1i, j≤2n+12 j

∏⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

A n( )

AHT 2n( ) = −3( )n n−1( )/2 3 j −i( ) + 2j −i +ni, j

∏⎛

⎝⎜⎞

⎠⎟A n( )

AQT 4n( ) =AHT 2n( )⋅A n( )2

Kuperberg, 2001: proved formulas for counting some new six-vertex models:

AUU 2n( ) = −3( )n2

22n 3 j −i( ) + 2j −i + 2n+11≤i, j≤2n+1

2 j

∏

1 −1 0 10 1 −1 00 0 0 00 0 1 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Kuperberg, 2001: proved formulas for many symmetry classes of ASM’s and some new ones

1 −1 0 10 1 −1 00 0 0 00 0 1 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

AUU 2n( ) = −3( )n2

22n 3 j −i( ) + 2j −i + 2n+11≤i, j≤2n+1

2 j

∏

Soichi Okada, Nagoya University

1993, Okada finds the equivalent of the λ-determinant for the other Weyl Denominator Formulas.

2004, Okada shows that the formulas for counting ASM’s, including those subject to symmetry conditions, are simply the dimensions of certain irreducible representations, i.e. specializations of Weyl Character formulas.

sλ 1,1,K ,1( ) =ρ + λ( )⋅rρ ⋅rr∈A2n−1

+∏ =

3 j +1( )2

⎢⎣⎢

⎥⎦⎥− 3i +1( )

2⎢⎣⎢

⎥⎦⎥

j −i1≤i< j≤2n∏

3−n(n−1)/2sλ 1,1,K ,1( ) =3i +1( )!n+ i( )!i=0

n−1

∏

Number of n n ASM’s is 3–n(n–1)/2

times the dimension of the irreducible representation of GL2n indexed by λ = n −1,n −1,n − 2,n − 2,K ,1,1,0,0( )

A2n−1+ = ei −ej 1≤i < j ≤2n{ }

ρ = n−12,n−3

2,K ,−n+ 12( )

dim Sp4n λ( ) =ρC + λ( )⋅rρC ⋅rr∈C2n

+∏

=6n+ 2 −

3i +12

⎢⎣⎢

⎥⎦⎥−

3 j +12

⎢⎣⎢

⎥⎦⎥

4n+ 2 −i − j

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟1≤i< j≤2n

∏3 j +12

⎢⎣⎢

⎥⎦⎥−

3i +12

⎢⎣⎢

⎥⎦⎥

j −i

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

3n+1−3i +12

⎢⎣⎢

⎥⎦⎥

2n+1−ii=1

2n

∏

Number of (2n+1) (2n+1) vertically symmetric ASM’s is 3–

n(n–1) times the dimension of the irreducible representation of Sp4n indexed by λ = n −1,n −1,n − 2,n − 2,K ,1,1,0,0( )

C2n+ = ei ±ej 1≤i < j ≤2n{ } U 2ei 1≤i ≤2n{ }

ρC =12

r =r∈C2n

+∑ 2n,2n−1,K ,1( )

NEW for 2004:

Number of (4n+1) (4n+1) vertically and horizontally symmetric ASM’s is 2–2n 3–n(2n–1)

times

λ = n −1,n −1,n − 2,n − 2,K ,1,1,0,0( )

μ = n − 12,n − 3

2,n − 32,K , 3

2, 32, 1

2( )

dim Sp4n λ( )×dim %O4n μ( ) =

ρC + λ( )⋅rρC ⋅rr∈C2n

+∏

⎛

⎝⎜

⎞

⎠⎟

ρD + μ( )⋅rρD ⋅rr∈D2n

+∏

⎛

⎝⎜

⎞

⎠⎟

C2n+ = ei ±ej 1≤i < j ≤2n{ } U 2ei 1≤i ≤2n{ }

ρC = 2n,2n−1,K ,1( )

D2n+ = ei ±ej 1≤i < j ≤2n{ }

ρD = 2n−1,2n−2,K ,1,0( )

Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture

Cambridge University Press & MAA, 1999

OKADA, Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups, arXiv:math.CO/0408234 v1 18 Aug 2004