Alternating sign matrices and descending plane partitions

Roger Behrend (Cardiff University)

Tuesday 13th November, 2012 15:00-16:00 Maths 325


Alternating sign matrices (ASMs) and descending plane partitions (DPPs) are combinatorial objects, each of which arose about 30 years ago, but in very different contexts.  However, it was conjectured by Mills, Robbins and Rumsey in 1983 that certain finite sets of ASMs have the same sizes as certain finite sets of DPPs, where these sets are comprised of all ASMs or DPPs with fixed values of particular statistics.  In this talk, the general background on ASMs and DPPs will be reviewed, and some details of a recent proof of the Mills-Robbins-Rumsey conjecture will be presented.  The proof will involve various standard results and techniques in integrable statistical mechanical models and combinatorics: a bijection between ASMs and configurations of the six-vertex model with domain-wall boundary conditions, the Izergin-Korepin determinant formula, a bijection between DPPs and certain sets of nonintersecting lattice paths, the Lindstrom-Gessel-Viennot theorem, and transformations of determinants using generating functions.  This is joint work with Philippe Di Francesco and Paul Zinn-Justin.

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