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Symmetry classes of alternating-sign matrices and square ice

Mathematical Physics & Probability

Speaker: Greg Kuperberg, UC Davis Mathematics
Location: 693 Kerr
Start time: Tue, Dec 5 2000, 3:10PM

Alternating-sign matrices are generalizations of permutation matrices first arose in certain algorithms for computing the determinant of a matrix. They came with an explicit product formula for their number. The correctness of this formula was a well-known conjecture in enumerative combinatorics, first proved in 1995 by Doron Zeilberger. Later that year I found another proof using the Izergin-Korepin determinant formula for a partition function for square ice, which in turn uses the Yang-Baxter equation. To combinatorialists, the Izergin-Korepin determinant was a surprising tool that changed the alternating-sign matrix question. To mathematical physicists, enumeration of alternating-sign matrices was a new application of square ice, or equivalently the six-vertex model.

Recently I discovered that the same methods apply to many symmetry classes of alternating-sign matrices, beginning with Izergin-Korepin-type determinants or Pfaffians and ending with product formulas. I will discuss the alternating-sign matrix story in both the old case and in the new ones.