Sign Matrix Polytopes

Abstract

Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, several new families of polytopes are defined as convex hulls of sign matrices, which are certain {0,1,-1}--matrices in bijection with semistandard Young tableaux. This bijection is refined to include standard Young tableau of certain shapes. One such shape is counted by the Catalan numbers, and the convex hull of these standard Young tableaux form a Catalan polytope. This Catalan polytope is shown to be integrally equivalent to the order polytope of the triangular poset: therefore the Ehrhart polynomial and volume can be combinatorial interpreted. Various properties of all of these polytope families are investigated, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes, and transportation polytopes.