Higher spin alternating sign matrices

Behrend, Roger E. ORCID: https://orcid.org/0000-0002-6143-7439 and Knight, Vincent Anthony ORCID: https://orcid.org/0000-0002-4245-0638 2007. Higher spin alternating sign matrices. Electronic Journal of Combinatorics 14 (1) , R83.

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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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	alternating sign matrix; semimagic square; convex polytope; higher spin vertex model
Additional Information:	38 pp.
Publisher:	Electronic Journal of Combinatorics
ISSN:	1077-8926
Last Modified:	05 May 2023 13:14
URI:	https://orca.cardiff.ac.uk/id/eprint/1730

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