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Equivariant Hilbert and Ehrhart series under translative group actions

Alessio D'Alì, Emanuele Delucchi

TL;DR

The paper develops a unified framework for equivariant Hilbert and Ehrhart theory under finite group actions that are translative, connecting CM Stanley–Reisner rings to unimodular triangulations of lattice polytopes. By combining a Cohen–Macaulay parameter approach with a colorful, coloring-preserving approach, it proves that equivariant Hilbert series have rational expressions with effective numerators and denominators of the form $(1-t)^{d+1}$, and that equivariant Ehrhart series of Polytopes with translative actions equal the corresponding equivariant Hilbert series of their triangulations. These results yield explicit formulas for order polytopes and alcoved polytopes, showing all entries of the equivariant $h$-vector and flag $h$-vector are effective under translative or balanced actions, and enabling practical computations in poset- and Coxeter-theoretic settings. The framework provides a toolkit for computing and understanding representation-theoretic structure in lattice-point enumeration and face-ring contexts, with concrete applications to posets and their associated polytopes.

Abstract

We study representations of finite groups on Stanley--Reisner rings of simplicial complexes and on lattice points in lattice polytopes. The framework of translative group actions allows us to use the theory of proper colorings of simplicial complexes without requiring an explicit coloring to be given. We prove that the equivariant Hilbert series of a Cohen--Macaulay simplicial complex under a translative group action admits a rational expression whose numerator is a positive integer combination of irreducible characters. This implies an analogous rational expression for the equivariant Ehrhart series of a lattice polytope with a unimodular triangulation that is invariant under a translative group action. As an application, we study the equivariant Ehrhart series of alcoved polytopes in the sense of Lam and Postnikov and derive explicit results in the case of order polytopes and of Lipschitz poset polytopes.

Equivariant Hilbert and Ehrhart series under translative group actions

TL;DR

The paper develops a unified framework for equivariant Hilbert and Ehrhart theory under finite group actions that are translative, connecting CM Stanley–Reisner rings to unimodular triangulations of lattice polytopes. By combining a Cohen–Macaulay parameter approach with a colorful, coloring-preserving approach, it proves that equivariant Hilbert series have rational expressions with effective numerators and denominators of the form , and that equivariant Ehrhart series of Polytopes with translative actions equal the corresponding equivariant Hilbert series of their triangulations. These results yield explicit formulas for order polytopes and alcoved polytopes, showing all entries of the equivariant -vector and flag -vector are effective under translative or balanced actions, and enabling practical computations in poset- and Coxeter-theoretic settings. The framework provides a toolkit for computing and understanding representation-theoretic structure in lattice-point enumeration and face-ring contexts, with concrete applications to posets and their associated polytopes.

Abstract

We study representations of finite groups on Stanley--Reisner rings of simplicial complexes and on lattice points in lattice polytopes. The framework of translative group actions allows us to use the theory of proper colorings of simplicial complexes without requiring an explicit coloring to be given. We prove that the equivariant Hilbert series of a Cohen--Macaulay simplicial complex under a translative group action admits a rational expression whose numerator is a positive integer combination of irreducible characters. This implies an analogous rational expression for the equivariant Ehrhart series of a lattice polytope with a unimodular triangulation that is invariant under a translative group action. As an application, we study the equivariant Ehrhart series of alcoved polytopes in the sense of Lam and Postnikov and derive explicit results in the case of order polytopes and of Lipschitz poset polytopes.
Paper Structure (16 sections, 32 theorems, 74 equations, 3 figures)

This paper contains 16 sections, 32 theorems, 74 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Lattice polytopes, simplicial complexes and group actions
  3. Definitions
  4. Group actions
  5. Translative and proper actions
  6. Background on equivariant Hilbert series
  7. Representations of finite groups
  8. The equivariant Hilbert series of a Cohen--Macaulay standard graded algebra
  9. Equivariant Hilbert series under translative actions
  10. The Cohen--Macaulay approach
  11. The colorful approach
  12. An application to equivariant Ehrhart series
  13. The equivariant Ehrhart series of a lattice polytope
  14. Main theorem
  15. Order polytopes
  16. ...and 1 more sections

Key Result

Theorem 1

Let $P$ be a lattice polytope and let $\Sigma$ be a unimodular lattice triangulation of $P$. Then

Figures (3)

  • Figure 1: The "radio-tower poset" $\mathsf{Rad}_k$ (left-hand side) and its poset of ideals $J(\mathsf{Rad}_k)$ (right-hand side).
  • Figure 2: The poset $T_n$.
  • Figure 3: The poset $Y$ for \ref{['ex:lips']}.

Theorems & Definitions (86)

  • Theorem : Betke--McMullen BetkeMcMullen
  • Theorem A: see \ref{['prop:translativity via lsop']} below
  • Theorem B: see \ref{['thm:EBM']} below
  • Theorem C: see \ref{['thm:orderp']} below for a more detailed statement
  • Theorem D: see \ref{['thm:Walcove']} below for a more detailed statement
  • Definition 2.1
  • Definition 2.2: Translative and proper actions
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 76 more