Higher SPIN alternating sign matrices

Roger E. Behrend ; Vincent A. Knight

The electronic journal of combinatorics, Tome 14 (2007)

Résumé

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 $r$.

Détail

DOI : 10.37236/1001

Classification : 05A15, 05B20, 52B05, 52B11, 82B20, 82B23