1526 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 Mx1 x2 L xn1 1 L 1

1 I xin i

i1

n

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 x j xi x j i j

x1n 1 x2

n 1 L xnn 1

M M O Mx1 x2 L xn1 1 L 1

1 I xin i

i1

n

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 x j xi x j 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 Mx1 x2 L xn1 1 L 1

1 I xin i

i1

n

Sn

Cauchy 1815

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

x1, x2 ,K , xn

x11 n 1 x2

1 n 1 L xn1 n 1

M M O Mx1

n 1 1 x2n 1 1 L xn

n 1 1

x1n x2

n L xnn

x1n 1 x2

n 1 L xnn 1

M M O Mx1

1 x21 L xn

1

x10 x2

0 L xn0

s 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 .

x11 n 1 x2

1 n 1 L xn1 n 1

M M O Mx1

n 1 1 x2n 1 1 L xn

n 1 1

x1n x2

n L xnn

x1n 1 x2

n 1 L xnn 1

M M O Mx1

1 x21 L xn

1

x10 x2

0 L xn0

s x1, x2 ,K , xn

Issai Schur (1875–1941)

s 1,1,K ,1 is the dimension of the representation

s 1,1,K ,1 r

rrAn 1

where n 1

2,n 3

2,K ,

1 n2

12

rrAn 1

,

1,2 ,K ,n ,An 1

ei e j 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 e j 1 i j n

Weyl 1939 The Classical Groups: their invariants and representations

Sp2n ; rx

x11 n x1

1 n L xn1 n xn

1 n

M O Mx1

n 1 x1 n 1 L xn

n 1 xn n 1

x1n x1

n L xnn xn

n

M O Mx1

1 x1 1 L xn

1 xn 1

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

Hermann Weyl (1885–1955)

Sp2n ;1r

r rrCn

where n,n 1,K ,1 12

rrCn

,

1,2 ,K ,n ,Cn

ei e j 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 Mx1

1 x1 1 L xn

1 xn 1

x1x2L xn n

xi x j 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 Mx1

1 x1 1 L xn

1 xn 1

x1x2L xn n

xi x j i j

xi2 1

i xix j 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

x1n 1

2 x1 n 1

2 L xnn 1

2 xn n 1

2

M O Mx1

12 x1

12 L xn

12 xn

12

x1x2L xn n 12

xi x j i j

xi 1 i

xix j 1 i j

x1n 1 x1

n1 L xnn 1 xn

n1

M O Mx1

0 x1 0 L xn

0 xn 0

x1x2L xn n 1

xi x j i j

2 xix j 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 detM nn detM n

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 detM nn detM n

1 detM1n

detM1,n1,n

det M detM1

1 detM nn detM n

1 detM1n

detM1,n1,n

det 1M detM

det a ji 1 i, j1

n ai a j

1i jn

David Robbins (1942–2003)

detM detM1

1 detM nn detM n

1 detM1n

detM1,n1,n

det M detM1

1 detM nn detM n

1 detM1n

detM1,n1,n

det

a bc d

ad bc

det

a b cd e fg h j

aej bdj afh 2 bfg cdh 3ceg

1 bde 1 fh

det

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

a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number

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

3 1 1 2a1,2a2,1a2,2

1 a2,3a3,2a3,3 1a3,4a4,3 L

det

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

a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number

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

3 1 1 2a1,2a2,1a2,2

1 a2,3a3,2a3,3 1a3,4a4,3 L

0 1 0 01 1 1 00 1 1 10 0 1 0

0 1 0 01 1 1 00 0 0 10 1 0 0

det

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

a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number

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

3 1 1 2a1,2a2,1a2,2

1 a2,3a3,2a3,3 1a3,4a4,3 L

0 1 0 01 1 1 00 1 1 10 0 1 0

0 1 0 01 1 1 00 0 0 10 1 0 0

det

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

a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number

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

3 1 1 2a1,2a2,1a2,2

1 a2,3a3,2a3,3 1a3,4a4,3 L

det xi, j Inv A

A ai , j 1 1 N A

xi, jai , j

i, j

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

det

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

a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number

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

3 1 1 2a1,2a2,1a2,2

1 a2,3a3,2a3,3 1a3,4a4,3 L

det xi, j Inv A

A ai , j 1 1 N A

xi, jai , j

i, j

xi x j Inv A 1 1 N A

x jn i ai , j

i, j

AAn

1i jn

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 00 ? ? ? ?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,k1

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,k1

n 2k 1

n 1k 1

n 2n k 1

n 1n k 1

An

3 j 1 !n j !j0

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 !j0

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 !j0

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

AAn

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

AAn

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 x j Inv A 1 1 N A

x jn i ai , j

i, j

AAn

1i jn

A n;0 n!

A n;1 An 3i 1 !n i !i0

n 1

A n;2 2n(n 1)/2

A n; x xN A

AAn

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 1

3 j i 1i, jnj i odd

Conjecture:

(MRR, 1983)

A n;0 n!

A n;1 An 3i 1 !n i !i0

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 !j0

n 1

AV 2n 1 3 n2 3 j i 1

j i 2n 11i, j2n12 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’sQuarter-turn symmetric ASM’s

December, 1992

Zeilberger announces a proof that # of ASM’s equals

3 j 1 !n j !j0

n 1

Doron Zeilberger

Rutgers University

December, 1992

Zeilberger announces a proof that # of ASM’s equals

3 j 1 !n j !j0

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,k1

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 IzerginVladimir Korepin

SUNY Stony Brook

1980’s

det1

xi y j axi y j

xi y j axi y j i, j1

nxi x j yi y j 1i jn

1 a 2N A an(n 1)/2 Inv A

AAn

xivert y j axi y j

SW, NE xi y j

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

AAn

det1

xi y j axi y j

xi y j axi y j i, j1

nxi x j yi y j 1i jn

1 a 2N A an(n 1)/2 Inv A

AAn

xivert y j axi y j

SW, NE xi y j

NW, SE

det1

xi y j axi y j

xi y j axi y j i, j1

nxi x j yi y j 1i jn

1 a 2N A an(n 1)/2 Inv A

AAn

xivert y j axi y j

SW, NE xi y j

NW, SE

z e i / 3 : RHS 3 n n 1 /2 An ,

z e i / 4 : RHS 2 n n 1 /2 2N A

AAn ,

z e i /6 : RHS 1 n n 1 /2 3N A

AAn .

a z 4 , xi z2 , yi 1

RHS z z 1 n n 1 z z 1 2N A

AAn

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, j2n12 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 11i, j2n1

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 11i, j2n1

2 j

Soichi Okada, Nagoya University

1993, Okada finds the equivalent of the -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.

s 1,1,K ,1 r

rrA2n 1

3 j 1 2

3i 1 2

j i1i j2n

3 n(n 1)/2 s 1,1,K ,1 3i 1 !n i !i0

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 n 1,n 1,n 2,n 2,K ,1,1,0,0

A2n 1 ei e j 1 i j 2n n 1

2,n 32,K , n 1

2

dim Sp4n C r

C rrC2n

6n 2

3i 12

3 j 1

2

4n 2 i j

1i j2n

3 j 1

2

3i 1

2

j i

3n 1 3i 1

2

2n 1 ii1

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 n 1,n 1,n 2,n 2,K ,1,1,0,0

C2n ei e j 1 i j 2n U 2ei 1 i 2n C

12

r rC2n

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

n 1,n 1,n 2,n 2,K ,1,1,0,0 n 1

2,n 32,n 3

2,K , 32, 3

2, 12

dim Sp4n dim %O4n

C rC rrC2n

D rD rrD2n

C2n ei e j 1 i j 2n U 2ei 1 i 2n C 2n,2n 1,K ,1

D2n ei e j 1 i j 2n D 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