Polytopes of alternating sign matrices with dihedral-subgroup symmetry

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 Chvátal--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.

Theorems & Definitions (84)

• Theorem 1.1: Edmonds edmonds1970submodular
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Edmonds and Johnson edmonds2003matching
• Lemma 1.5
• Theorem 1.6
• Theorem 1.7
• proof
• Theorem 2.1: Behrend and Knight behrend2007higher, and Striker striker2007alternatingstriker2009alternating
• Theorem 2.2: Behrend and Knight behrend2007higher, and Striker striker2007alternatingstriker2009alternating