Diagonally and antidiagonally symmetric alternating sign matrices

Roger Behrend (Cardiff)

Thursday 18th February, 2016 16:00-17:00 Maths 522


An alternating sign matrix (ASM) is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1.  It was conjectured by Mills, Robbins and Rumsey in 1982 that the number of ASMs of fixed size is given by a certain simple product formula.  A relatively short proof of this conjecture was obtained by Kuperberg in 1996, using the Izergin-Korepin determinant formula for the partition function of the six-vertex model on a square grid with domain-wall boundary conditions, together with a bijection between ASMs and configurations of that model.  It was also conjectured by Robbins in the mid 1980's that the number of ASMs of fixed odd size which are invariant under diagonal and antidiagonal reflection is given by a simple product formula.  This conjecture has only recently been proved, in joint work with Ilse Fischer and Matjaz Konvalinka (see arXiv:1512.06030).  Our proof again uses connections with a particular case of the six-vertex model.  In the first part of this talk, I'll introduce ASMs, and review Kuperberg's proof.  In the second part, I'll outline the proof of the conjecture for the enumeration of diagonally and antidiagonally symmetric ASMs, and present some new results for the associated case of the six-vertex model.

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