The equivariant Ehrhart theory of polytopes with order-two symmetries

TL;DR

This paper explores the equivariant Ehrhart theory of polytopes with order-two symmetries, analyzing effectiveness and providing counterexamples that challenge existing conjectures, including cases with rational coordinates.

Contribution

It introduces new insights into the effectiveness of equivariant $H^*$-polynomials for symmetric polytopes and presents counterexamples involving rational coordinates.

Findings

Counterexample to effectiveness conjecture with rational vertices

Symmetric edge polytopes of cycles analyzed

Ehrhart function with period one matching lattice polytope

Abstract

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant $H^*$-polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycles and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope.