David M. Bressoud Macalester College, St. Paul, MN Talk given at University of Florida October 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

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

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

Cauchy 1815 Any alternating function in divided by the Vandermonde determinant yields a symmetric function:

Cauchy 1815 Any alternating function in divided by the Vandermonde determinant yields a symmetric function: Called the Schur function. I.J. Schur (1917) recognized it as the character of the irreducible representation of GL n indexed by. Issai Schur (1875– 1941)

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

Weyl 1939 The Classical Groups: their invariants and representations is the character of the irreducible representation, indexed by the partition, of the symplectic group (the subgoup of GL 2n of isometries). Hermann Weyl (1885–1955)

The dimension of the representation is

Weyl 1939 The Classical Groups: their invariants and representations is a symmetric polynomial. As a polynomial in x 1 it has degree n + 1 and roots at

Weyl 1939 The Classical Groups: their invariants and representations is a symmetric polynomial. As a polynomial in x 1 it has degree n + 1 and roots at

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

Desnanot-Jacobi adjoint matrix thereom (Desnanot for n ≤ 6 in 1819, Jacobi for general case in 1833 is matrix M with row i and column j removed. Given that the determinant of the empty matrix is 1 and the determinant of a 1  1 is the entry in that matrix, this uniquely defines the determinant for all square matrices. Carl Jacobi (1804–1851)

David Robbins (1942–2003)

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

n n A n = 2  3  7 = 3  11  13 = 2 2  11  13 2 = 2 2  13 2  17  19 = 2 3  13  17 2  19 2 = 2 2  5  17 2  19 3  23 How many n  n alternating sign matrices?

n n A n = 2  3  7 = 3  11  13 = 2 2  11  13 2 = 2 2  13 2  17  19 = 2 3  13  17 2  19 2 = 2 2  5  17 2  19 3  23

n n A n There is exactly one 1 in the first row

n n A n … There is exactly one 1 in the first row

1 1 2/ /3 3 3/ / / / / / /2 429

1 1 2/ /3 3 3/ /4 14 5/5 14 4/ / / / / / / / / /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

Numerators:

Conjecture 1: Numerators:

Conjecture 1: Conjecture 2 (corollary of Conjecture 1):

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

Conjecture 2 (corollary of Conjecture 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

Conjecture: (MRR, 1983)

Mills & Robbins (suggested by Richard Stanley) (1991) Symmetries of ASM’s 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 Doron Zeilberger Rutgers University

December, 1992 Zeilberger announces a proof that # of ASM’s equals 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:

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 Notices Physicists have been studying ASM’s for decades, only they call them square ice (aka the six-vertex model ) 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 vertical I = inversion number = N + # of SW x 2, y 3

Anatoli Izergin Vladimir Korepin SUNY Stony Brook 1980’s

Proof: LHS is symmetric polynomial in x’s and in y’s Degree n – 1 in x 1 By induction, LHS = RHS when x 1 = y 1 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 x 1 By induction, LHS = RHS when x 1 = –y 1 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

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:

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

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

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.

Number of n  n ASM’s is 3 –n(n–1)/2 times the dimension of the irreducible representation of GL 2n indexed by

Number of (2n+1)  (2n+1) vertically symmetric ASM’s is 3 –n(n–1) times the dimension of the irreducible representation of Sp 4n indexed by

NEW for 2004: Number of (4n+1)  (4n+1) vertically and horizontally symmetric ASM’s is 2 –2n 3 –n(2n–1) times

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/ v1 18 Aug 2004