Alternating sign matrices, extensions and related cones

Abstract

An alternating sign matrix , or ASM, is a (0,±1)(0,±1)-matrix where the nonzero entries in each row and column alternate in sign, and where each row and column sum is 1. We study the convex cone generated by ASMs of order n , called the ASM cone, as well as several related cones and polytopes. Some decomposition results are shown, and we find a minimal Hilbert basis of the ASM cone. The notion of (±1)(±1)-doubly stochastic matrices and a generalization of ASMs are introduced and various properties are shown. For instance, we give a new short proof of the linear characterization of the ASM polytope, in fact for a more general polytope. Finally, we investigate faces of the ASM polytope, in particular edges associated with permutation matrices.