Polytopes of alternating sign matrices with dihedral-subgroup symmetry

TL;DR

This paper studies the convex polytopes of symmetric alternating sign matrices, providing explicit descriptions for most symmetry classes and developing a core-assembly framework that simplifies polyhedral analysis.

Contribution

It introduces a uniform core-assembly framework for analyzing symmetry classes of ASMs, enabling explicit polyhedral descriptions and efficient algorithms.

Findings

Explicit polynomial-size linear inequalities for most symmetry classes.

Dimension and facet descriptions for these polytopes.

Structured inequalities needed for quarter-turn symmetry class.

Abstract

We investigate the convex hulls of the eight dihedral symmetry classes of $n \times n$ alternating sign matrices, i.e., ASMs invariant under a subgroup of the symmetry group of the square. Extending the prefix-sum description of the ASM polytope, we develop a uniform core--assembly framework: each symmetry class is encoded by a set of core positions and an affine assembly map that reconstructs the full matrix from its core. This reduction transfers polyhedral questions to lower-dimensional core polytopes, which are better suited to the tool set of polyhedral combinatorics, while retaining complete information about the original symmetry class. For the vertical, vertical--horizontal, half-turn, diagonal, diagonal--antidiagonal, and total symmetry classes, we give explicit polynomial-size linear inequality descriptions of the associated polytopes. In these cases, we also determine the dimension and provide facet descriptions. The quarter-turn symmetry class behaves differently: the natural relaxation admits fractional vertices, and we need to extend the system with a structured family of parity-type Chv\'atal--Gomory inequalities to obtain the quarter-turn symmetric ASM polytope. Our framework leads to efficient algorithms for computing minimum-cost ASMs in each symmetry class and provides a direct link between the combinatorics of symmetric ASMs and tools from polyhedral combinatorics and combinatorial optimization.