### The volume and the Ehrhart polynomial of the alternating sign matrix polytope

Combinatorics Seminar

5th May 2020, 11:00 am – 12:00 pm

Fry Building, 2.04

An alternating sign matrix of order n, or an ASM for short, is an n by n matrix with entries from {-1,0,1} such that all row and column sums equal 1 and along each row and column the non-zero entries alternate in sign. The set of all ASMs of order n is denoted by ASM(n). They were first introduced by Bill Mills, Dave Robbins and Howard Rumsey in 1982. It turns out that the ASMs have many faces and that have connections with many other combinatorial objects such as plane partitions, six-vertex configurations and so forth. In this talk I will discuss some new results for the volume and Ehrhart polynomial of the ASM polytope by constructing an explicit bijection between higher spin ASMs and a disjoint union of sets of certain (P,W)-partitions (where P is a certain partially ordered set and W is a labelling). Higher spin ASMs are natural generalizations of ASMs introduced by Behrend and Knight (They are n by n matrices with integer entries such that each row and column sum is equal to the non-negative integer r, and such that partial line sums extending from either end along each row and column are non-negative). Moreover, a formula is derived for the number of higher spin ASMs, or equivalently for the Ehrhart polynomial of the ASM polytope. The relative volume of the ASM polytope is then given by the leading term of its Ehrhart polynomial. Evaluation of the formula involves computing numbers of linear extensions of P, and numbers of descents in these linear extensions. Details of this computation are presented for the cases of the ASM polytope of order 4, 5, 6 and 7.

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