Higher Spin Alternating Sign Matrices

Abstract

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The case r=1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension (n-1)^2 whose vertices are the standard alternating sign matrices of size n. It then follows that, for fixed n, these matrices are enumerated by an Ehrhart polynomial in r.

41 pages; v2: minor changes

References in corpus (6)

• Symmetry Classes of Alternating Sign Matrices
• Totally Symmetric Self-Complementary Plane Partitions and Quantum Knizhnik-Zamolodchikov equation: a conjecture
• Combinatorial point for higher spin loop models
• On refined enumerations of totally symmetric self-complementary plane partitions I
• The alternating sign matrix polytope
• Osculating Paths and Oscillating Tableaux