Sign-restricted matrices of $0$'s, $1$'s, and $-1$'s

TL;DR

This paper introduces and analyzes sign-restricted matrices (SRMs), a generalization of alternating sign matrices, focusing on their structure, maximum nonzeros, sum vectors, and related algebraic and geometric properties.

Contribution

It characterizes SRMs' maximum nonzeros, sum vectors, and extends algebraic structures like Bruhat order to these matrices, providing new insights into their combinatorial and geometric properties.

Findings

Maximum number of nonzeros in SRMs determined

Characterization of possible row and column sum vectors

Extension of Bruhat order to SRMs and related polytopes

Abstract

We study {\em sign-restricted matrices} (SRMs), a class of rectangular $(0, \pm 1)$-matrices generalizing the alternating sign matrices (ASMs). In an SRM each partial column sum, starting from row 1, equals 0 or 1, and each partial row sum, starting from column 1, is nonnegative. We determine the maximum number of nonzeros in SRMs and characterize the possible row and column sum vectors. Moreover, a number of results on interchange operations are shown, both for SRMs and, more generally, for $(0, \pm 1)$-matrices. The Bruhat order on ASMs can be extended to SRMs with the result a distributive lattice. Also, we study polytopes associated with SRMs and some relates decompositions.