The alternating sign matrix polytope

TL;DR

This paper introduces the alternating sign matrix polytope, characterizes its structure through inequalities, and explores its geometric properties, including facets, vertices, and face lattice, linking it to square ice configurations.

Contribution

It provides the first convex polytope description of alternating sign matrices, extending classical results on permutation matrices and doubly stochastic matrices.

Findings

Counted facets and vertices of the polytope

Described the projection to the permutohedron

Characterized the face lattice via square ice configurations

Abstract

We define the alternating sign matrix polytope as the convex hull of nxn alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give a complete characterization of its face lattice in terms of modified square ice configurations. Furthermore we prove that the dimension of any face can be easily determined from this characterization.