Higher Spin Alternating Sign Matrices

TL;DR

This paper introduces higher spin alternating sign matrices, generalizing standard alternating sign matrices and semimagic squares, and explores their geometric and combinatorial properties, including enumeration via Ehrhart polynomials.

Contribution

The paper defines higher spin alternating sign matrices and connects them to convex polytopes, providing a geometric framework and enumeration method.

Findings

Higher spin alternating sign matrices correspond to integer points in dilated convex polytopes.

Enumeration of these matrices is given by an Ehrhart polynomial in the parameter r.

The matrices generalize known classes like standard alternating sign matrices and semimagic squares.

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.