Higher Spin Alternating Sign Matrices

Details Higher Spin Alternating Sign Matrices Higher Spin Alternating Sign Matrices

2007-08-20

arxiv.org/abs/0708.2522v2

Electron. J. Combin. 14 (2007) Research Paper 83

Mathematics

Combinatorics

41 pages; v2: minor changes

Scientific paper

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.

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