Equivariant Ehrhart theory

TL;DR

This paper develops an equivariant extension of Ehrhart theory, connecting lattice point enumeration with representation theory, and applies it to rational and symmetric polytopes, including a Weyl group action character formula.

Contribution

It introduces an equivariant Ehrhart theory, generalizing classical results and linking lattice point enumeration with representation theory and geometry.

Findings

Proves representation-theoretic analogues of classical Ehrhart results

Applies theory to rational and symmetric polytopes

Recovers a Weyl group action character formula

Abstract

Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of numerous classical results, and give applications to the Ehrhart theory of rational polytopes and centrally symmetric polytopes. We also recover a character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of a Weyl group on the cohomology of a toric variety associated to a root system.