Graded Ehrhart Theory of Unimodular Zonotopes

TL;DR

This paper develops a graded Ehrhart theory for unimodular zonotopes, linking lattice point counts to Tutte polynomials, and explores the algebraic structure of associated harmonic algebras, proving several conjectures.

Contribution

It introduces a new graded Ehrhart theory for unimodular zonotopes, connecting it to matroid invariants and algebraic geometry, and classifies properties of their harmonic algebras.

Findings

Graded lattice point count equals a q-evaluation of the Tutte polynomial.

The harmonic algebra is finitely generated and Cohen–Macaulay.

Classification of Gorenstein harmonic algebras for unimodular zonotopes.

Abstract

Graded Ehrhart theory is a new $q$-analogue of Ehrhart theory based on the orbit harmonics method. We study the graded Ehrhart theory of unimodular zonotopes from a matroid-theoretic perspective. Generalizing a result of Stanley (1991), we prove that the graded lattice point count of a unimodular zonotope is a $q$-evaluation of its Tutte polynomial. We conclude that the graded Ehrhart series of a unimodular zonotope is rational and obeys graded Ehrhart--Macdonald reciprocity. In an algebraic direction, we prove that the harmonic algebra of a unimodular zonotope is a coordinate ring of its associated arrangement Schubert variety. Using the geometry of arrangement Schubert varieties, we prove that the harmonic algebra of a unimodular zonotope is finitely generated and Cohen--Macaulay. We also give an explicit presentation of the harmonic algebra of a unimodular zonotope in terms of generators and relations. We conclude by classifying which unimodular zonotopes have Gorenstein harmonic algebras. Our work answers, in the special case of unimodular zonotopes, two conjectures of Reiner and Rhoades (2024).