Enumerating alternating-sign matrices and their symmetry classes

This talk is a review of some old work in enumerative combinatorics that has also attracted interest in the mathematical physics community. In 1983, in an investigation of determinant formulas, David Robbins discovered the class of alternating-sign matrices (ASMs). These are 1,0,-1-valued matrices whose non-zero entries alternate in sign, and begin and end with 1, in every row and column. He conjectured a special formula for the number of ASMs of order n. The formula is a product of factorials divided by another product of factorials; in particular the number is a \round\ number with no large prime factors. Later he conjecture round formulas for many symmetry classes of ASMs, i.e., ASMs with an imposed symmetry. These formulas turned out to be easier to find (with computer experiments) than to prove. There are now several proofs of the formula for the original class of ASMs. The first proof, by Doron Zeilberger, was quite complicated. (Another recent proof is by Ilse Fischer.) I will describe a proof based on the Yang-Baxter equation, which also arises in knot theory in the study of topological invariants such as the Jones polynomial. This proof can be generalized to many symmetry classes of ASMs, and it led to a connection with Schur polynomials discovered by Soichi Okada. But, none of the proofs establish any bijections, even between types of ASMs that occur in equal numbers.