EXPLOITING SYMMETRIES: ALTERNATING SIGN

EXPLOITING SYMMETRIES: ALTERNATING SIGN MATRICES AND THE WEYL CHARACTER FORMULAS DAVID M. BRESSOUD

1. Introduction Symmetry is one of the great organizing principles of mathematics. If the purpose of mathematics is to discover the underlying patterns of our universe, then the search for symmetry indicates where to look for these patterns. Symmetry is also a powerful tool that provides us with simple and elegant proofs of algebraic results. At the heart of any argument involving symmetry lies the full symmetric group, Sn . The structure of the full symmetry group is intimately bound to the determinant. The purpose of this paper is to illustrate some of the power and beauty of determinant evaluations, beginning with Cauchy’s proof of the Vandermonde determinant evaluation and ending with the Izergin-Korepin determinant expansion for the six-vertex model with domain wall boundary conditions. In section 2, I will describe how symmetry leads to simple proofs of the Vandermonde determinant formula, and how this approach is generalized to the Weyl denominator formulas. Sections 3 through 5 introduce alternating sign matrices, explain their connection to the Vandermonde determinant, and describe some of the conjectures that arose between 1980 and 1995. In section 6, I will explain the connection between alternating sign matrices and the 6-vertex model of statistical mechanics, sketch the proof of the determinant formula for the partition function of the six-vertex model, and summarize recent work of Kuperberg and Okada that exploits the symmetry implicit in Baxter’s triangle-to-triangle relation. 2. The Vandermonde determinant and its successors The Vandermonde determinant Cauchy in 1815 [5], n−1 x1 xn−1 · · · xn−1 n 2 . . .. .. . . . . . . (1) 1 1 x1 x2 · · · x1n x0 x0 · · · x0 1

2

n

formula was first stated in its full generality by A.-L. n X Y Y n−σ(i) (−1)Inv(n) xi = (xi − xj ) . = i=1 1≤i