Symmetry classes of alternating-sign matrices

Greg Kuperberg

UC Davis

March 19,
refreshments at 3:45pm


Alternating-sign matrices form a well-known class of combinatorial objects that is worth enumerating. The matrices themselves and the conjecture for their number were discovered by Mills, Robbins and Rumsey; the conjecture was first proved by Doron Zeilberger. Shortly after Zeilberger's proof was confirmed, I found another proof based on the Yang-Baxter equation and the Izergin-Korepin determinant.

But this one conjecture is not the end of the story. Robbins (in one case, Mills) also studied symmetry classes of alternating-sign matrices, by analogy with symmetry classes of plane partitions. Many of these symmetry classes also have round (i.e. smooth) enumerations according to conjecture and computer experiments. I will describe my program to generalize the Izergin-Korepin determinant and its evaluation to these other symmetry classes. So far the method only enumerates some of the symmetry classes listed by Robbins, but it also leads to new classes of matrices that can also be enumerated. Moreover, alternating-sign matrices could be only the sl(2) case of constructions in quantum algebra that generalize to many complex simple Lie algebras.

Speaker's Contact Info: greg(at-sign)

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