Polytopes associated to Dihedral Groups

TL;DR

This paper studies the convex hull of permutation matrices representing dihedral symmetries, providing a complete facet description and exploring its algebraic and combinatorial properties.

Contribution

It offers a full facet description of the polytope associated with dihedral group symmetries and analyzes its Gorenstein and Ehrhart properties.

Findings

Complete facet description of the dihedral symmetry polytope

Identification of the polytope as Gorenstein

Determination of the Ehrhart h*-vector

Abstract

In this note we investigate the convex hull of those $n \times n$-permutation matrices that correspond to symmetries of a regular $n$-gon. We give the complete facet description. As an application, we show that this yields a Gorenstein polytope, and we determine the Ehrhart $h^*$-vector.