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A doubly-refined enumeration for alternating sign matrices


Speaker: Dan Romik, Einstein Institute of Mathematics, Univesity of Jerusalem
Location: 1147 MSB
Start time: Mon, Jan 12 2009, 4:10PM

Alternating-sign matrices (ASM's) are important objects in algebraic combinatorics and statistical physics, with connections to many subjects such as domino tilings, the square ice model, plane partitions and more. In 1982 Mills, Robbins and Rumsey made two beautiful conjectures on the total and refined enumeration of alternating sign matrices of given order N. The conjectures were proved by Doron Zeilberger in the 1990's, in the refined case building on the square ice methods introduced by Kuperberg. I will survey these results and describe a doubly-refined enumeration studied by myself and Ilse Fischer. Our results and conjectures point the way towards a much more general k-refined enumeration of ASM's based on their first k rows, which could be helpful in solving some important open problems on the behavior of uniformly random alternating sign matrices.