Techniques in equivariant Ehrhart theory

TL;DR

This paper reviews techniques in equivariant Ehrhart theory, focusing on methods like zonotopal decompositions and symmetric triangulations, and applies them to various polytopes to expand understanding of their symmetry-related lattice point enumeration.

Contribution

It provides a comprehensive catalogue of techniques for equivariant Ehrhart theory and demonstrates their application to new classes of polytopes.

Findings

Expanded the set of polytopes with known equivariant Ehrhart properties

Developed certificates for the existence of invariant hypersurfaces

Applied techniques to hypersimplices, orbit polytopes, and graphic zonotopes

Abstract

Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including zonotopal decompositions, symmetric triangulations, combinatorial interpretation of the $h^\ast$-polynomial, and certificates for the (non)existence of invariant non-degenerate hypersurfaces. We apply these methods to several families of examples including hypersimplices, orbit polytopes, and graphic zonotopes, expanding the library of polytopes for which their equivariant Ehrhart theory is known.