The characteristic quasi-polynomials of hyperplane arrangements with actions of finite groups

TL;DR

This paper introduces an equivariant characteristic quasi-polynomial for hyperplane arrangements with finite group actions, expressing it via induced characters and applying it to Coxeter arrangements, especially type A.

Contribution

It develops an equivariant version of characteristic quasi-polynomials, linking permutation characters with Ehrhart quasi-polynomials and providing explicit computations for Coxeter arrangements.

Findings

Permutation character is a quasi-polynomial in q.

The character can be expressed as a sum of induced characters.

Explicit formulas are provided for type A Coxeter arrangements.

Abstract

In this paper, we introduce an equivariant version of the characteristic quasi-polynomials as the permutation characters on the complement of mod $q$ hyperplane arrangements. We prove that the permutation character is a quasi-polynomial in $q$, and show that it can be expressed by the sum of the induced characters of an equivariant version of the Ehrhart quasi-polynomials. Furthermore, we consider the case of the Coxeter arrangements, and compute in detail for type $A_\ell$.