Sign matrix polytopes from Young tableaux

Abstract

Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, we define several new families of polytopes as convex hulls of sign matrices, which are certain {0, 1, −1}-matrices in bijection with semistandard Young tableaux. We investigate various properties of these polytopes, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes and transportation polytopes.2010 Mathematics Subject Classification. 05A05, 52B05.

Partial permutation and alternating sign matrix polytopes

“…Partial alternating sign matrices are a subset of sign matrices, which differ from Definition 2.5 in that each row partial sum is not restricted to {0, 1} as in (2.5), but may equal any non-negative integer. See [19] for information about polytopes whose vertices are sign matrices and Lemma 4.9 for the relationship between these polytopes.…”

“…Proof. In [19,Theorem 4.3], a hyperplane is constructed that separates a given m × n sign matrix from all other m × n sign matrices. Since m × n partial alternating sign matrices are a subset of m × n sign matrices, this hyperplane must separate a given m × n partial alternating sign matrix from all others.…”

“…In this paper, we define and study a more general class of polytopes composed as convex hulls of m × n partial alternating sign matrices, denoted PASM(m, n). We use machinery developed in this study of sign matrix polytopes [19] to determine the inequality descriptions of these polytopes, as well as facet enumerations and a description of their face lattices. Finally, we investigate the partial permutohedron, P(m, n), and show it is a projection of the polytopes from the first part of the paper.…”

Alternating sign and sign-restricted matrices: representations and partial orders

“…Matrices with constraints on the sign of entries, or the sum of entries, are of interest in many settings. A sign-restricted matrix, or SRM, is an m × n (0, ±1)-matrix A such that each partial column sum, starting from row 1, equals 0 or 1, and each partial row sum, starting from column 1, is nonnegative, see [11,17]. The constraint on each column implies that its nonzeros alternate, starting with a +1.…”

“…The constraint on each column implies that its nonzeros alternate, starting with a +1. This notion was introduced in [1] and it was shown that this class is in bijection with so-called semistandard Young tableaux (see also [17]). An important subclass of SRMs is the alternating sign matrices (ASMs); these are square SRMs where both the rows and columns are alternating starting and ending with a +1, see [2,3,6,8,9,10,18].…”