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Ehrhart Theory over Abelian Group Rings

Robert Davis, Jesús A. De Loera, Alexey Garber, Katharina Jochemko, Josephine Yu

TL;DR

This work generalizes Ehrhart theory by encoding lattice-point weights through a homomorphism $\varphi: \mathbb{Z}^d\to G$ into a group ring $R[G]$, producing $\varphi$-Ehrhart functions $\mathrm{ehr}(P;\varphi;n)$ and series $\mathcal{E}(P;\varphi;t)$ that encompass polynomial weights, Chapoton's $q$-weights, and equivariant variants. The authors establish rationality, reciprocity, and a $\varphi$-Brion decomposition for the weighted, group-ring setting, using a multivariate transform $\sigma(Q,\varphi;\mathbf{z})$ and cone decompositions to transfer classical Ehrhart results to $R[G]$-valued contexts. They show how classical results for lattice and rational polytopes emerge as special cases, and develop weighted extensions, including $\varphi$-weighted, $w$-weighted, and equivariant theories, with explicit examples. The framework provides a unified approach to a broad spectrum of weighted lattice-point enumerations and suggests future directions in computation, modular counting, and connections to harmonic and graded Ehrhart theory.

Abstract

We study Ehrhart series with coefficients in Abelian group rings. This opens new enumeration applications and unifies earlier variants, in particular, polynomial weighted, $q$-weighted, and equivariant Ehrhart series.

Ehrhart Theory over Abelian Group Rings

TL;DR

This work generalizes Ehrhart theory by encoding lattice-point weights through a homomorphism into a group ring , producing -Ehrhart functions and series that encompass polynomial weights, Chapoton's -weights, and equivariant variants. The authors establish rationality, reciprocity, and a -Brion decomposition for the weighted, group-ring setting, using a multivariate transform and cone decompositions to transfer classical Ehrhart results to -valued contexts. They show how classical results for lattice and rational polytopes emerge as special cases, and develop weighted extensions, including -weighted, -weighted, and equivariant theories, with explicit examples. The framework provides a unified approach to a broad spectrum of weighted lattice-point enumerations and suggests future directions in computation, modular counting, and connections to harmonic and graded Ehrhart theory.

Abstract

We study Ehrhart series with coefficients in Abelian group rings. This opens new enumeration applications and unifies earlier variants, in particular, polynomial weighted, -weighted, and equivariant Ehrhart series.

Paper Structure

This paper contains 8 sections, 8 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proofs of Rationality, Reciprocity, and $\varphi$-Brion
  3. Classical Ehrhart Results from $\varphi$-Ehrhart theory
  4. Weights in Lattice Points via $\varphi$-Ehrhart functions
  5. Chapoton's $q$-Ehrhart
  6. Polynomial Weights
  7. Equivariant $\varphi$-Ehrhart theory
  8. Conclusions

Key Result

Theorem 1.3

For a half-open rational simplicial cone $\mathcal{C}$, we have

Figures (1)

  • Figure 1: Left: The first dilations of the unit square in $\mathbb{Z}^2$. Each point $v\in\mathbb{Z}^2$ is labeled with the $\varphi(v)$ from Example \ref{['ex:first example']}. Right: Interpretations of $\mathop{\mathrm{ehr}}\nolimits(P,\varphi;n)$ for some pairs of $G$ and $R$. Each uses the homomorphism $\varphi: \mathbb{Z}^d \to G$ sending standard basis vectors to the same generator $\omega\in G$. We use $K_i$ to denote the coset $ie_1 + \ker(\varphi)$ of $\mathbb{Z}^d/\ker(\varphi)$.

Theorems & Definitions (19)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3: Rationality
  • Corollary 1.4
  • Example 1.5
  • Theorem 1.6: Reciprocity
  • Theorem 1.7: $\varphi$-Brion
  • proof : Proof of \ref{['lem:brion_group']} (Rationality)
  • Remark 2.1
  • proof : Proof of \ref{['thm:phi-reciprocity']} (Reciprocity)
  • ...and 9 more