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

Ryo Uchiumi

TL;DR

This work extends characteristic quasi-polynomials to an equivariant setting by introducing the permutation character on the mod-$q$ complement of a $oldsymbol{ m ho}$-invariant hyperplane arrangement. The authors prove that the permutation character $oldsymbol{\chi}_{\mathscr{A},q}$ is a gcd-property quasi-polynomial in $q$, and they express it as a sum of induced characters from stabilizers of chambers, linking to an equivariant Ehrhart framework. A torus/torsion-points viewpoint and the equivariant Ehrhart theory yield a cohesive decomposition, with concrete results for Coxeter arrangements and explicit type-$A_\ell$ cases where $oldsymbol{\chi}_{\mathscr{A},q} = \mathrm{Ind}^{W}_{W_{A_0}} \boldsymbol{\chi}_{A_0,q}$. These results bridge hyperplane arrangement combinatorics, toric geometry, and representation theory, providing computable formulas and insights for symmetry-structured settings.

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 $A_\ell$.

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

TL;DR

This work extends characteristic quasi-polynomials to an equivariant setting by introducing the permutation character on the mod- complement of a -invariant hyperplane arrangement. The authors prove that the permutation character is a gcd-property quasi-polynomial in , and they express it as a sum of induced characters from stabilizers of chambers, linking to an equivariant Ehrhart framework. A torus/torsion-points viewpoint and the equivariant Ehrhart theory yield a cohesive decomposition, with concrete results for Coxeter arrangements and explicit type- cases where . These results bridge hyperplane arrangement combinatorics, toric geometry, and representation theory, providing computable formulas and insights for symmetry-structured settings.

Abstract

In this paper, we introduce an equivariant version of the characteristic quasi-polynomials as the permutation characters on the complement of mod hyperplane arrangements. We prove that the permutation character is a quasi-polynomial in , 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 .

Paper Structure

This paper contains 14 sections, 28 theorems, 161 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quasi-polynomials
  3. Main Results
  4. Equivariant version of characteristic quasi-polynomials
  5. Group actions and representations
  6. Setting
  7. Quasi-polynomiality
  8. Periods and Examples
  9. Via $q$-torsion points of the torus
  10. Via the equivariant Ehrhart theory
  11. Coxeter arrangements with the Weyl group action
  12. Coxeter arrangements and Weyl groups
  13. Fundamental alcoves
  14. Case of type $A_\ell$

Key Result

Theorem 1.1

The permutation character $\chi_{\mathscr{A} ,q}$ is a quasi-polynomial in $q$. Furthermore, $\chi_{\mathscr{A} ,q}$ has the gcd-property.

Figures (1)

  • Figure 1: $\mathscr{A} ^{\mathop{\mathrm{aff}}\nolimits}$ in \ref{['egperiod10']} and \ref{['egdecomp']}, the points $\bullet$ are fixed by $\rho_T(\gamma)$, the points $\circ$ are fixed by $\rho_T(\gamma^2)$.

Theorems & Definitions (48)

  • Theorem 1.1: \ref{['Main result 1']}
  • Theorem 1.2: \ref{['Main result 2']}
  • Theorem 1.3: \ref{['weylver']}
  • Theorem 2.1: KamiyaTakemuraTerao
  • Proposition 1: Higashitani--Tran--Yoshinaga HigashitaniTranYoshinaga
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2.2: UchiumiYoshinaga
  • ...and 38 more