The quasi-polynomiality of mod q permutation representation for a linear finite group action on a lattice

TL;DR

This paper proves that the mod q permutation representation of a finite group acting linearly on a lattice is a quasi-polynomial in q and extends related results to mod q-analogues, including reciprocity for irreducible multiplicities.

Contribution

It introduces the quasi-polynomiality of mod q permutation representations and develops mod q-analogues of known Ehrhart results, including reciprocity for irreducible decompositions.

Findings

Permutation representations are quasi-polynomial in q.

Established mod q-analogues of Ehrhart quasi-polynomials.

Proved reciprocity results for multiplicities of irreducible components.

Abstract

For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod q-analogues of results by Stapledon for equivariant Ehrhart quasi-polynomials. We also prove a reciprocity-type result for multiplicities of irreducible decompositions.