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Graded Ehrhart Theory of Unimodular Zonotopes

Colin Crowley, Ethan Partida

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).

Graded Ehrhart Theory of Unimodular Zonotopes

Abstract

Graded Ehrhart theory is a new -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 -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).
Paper Structure (19 sections, 45 theorems, 125 equations, 2 figures)

This paper contains 19 sections, 45 theorems, 125 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Enumerative Results
  3. Algebraic Results
  4. Illustration of results
  5. Background
  6. Graded Ehrhart theory
  7. Matroids and the Tutte polynomial
  8. Zonotopes and zonotopal algebras
  9. Arrangement Schubert varieties
  10. Quantum integer-valued polynomials
  11. Graded Ehrhart polynomials and quantum reciprocity
  12. Quantum Reciprocity
  13. Graded Ehrhart Polynomials
  14. Harmonic algebras and arrangement Schubert varieties
  15. Gorenstein Harmonic Algebras
  16. ...and 4 more sections

Key Result

Proposition 1.1

Let $m\geq 1$ be an integer, $Z=A\cdot [0,1]^n\subseteq \mathbb{R}^d$ be a unimodular zonotope and $M$ be the matroid of $A$. We have that, where $T_M(x,y)$ is the Tutte polynomial of $M$ and $[k]_q := \frac{1-q^k}{1-q}$ is the $k$th $q$-integer.

Figures (2)

  • Figure 1: An example from the proof of \ref{['lem:reduce-to-linear']}, where $m = 3$, $n = 5$, $(i,j)=(2,3)$, and $C$ is indicated by the boxed entries. The red entries highlight which elements of the columns we've altered to construct $A'$. The matrix $A'$ has the property that its second row is equal to the set $(\mathop{\mathrm{\mathrm{colsupp}}}\nolimits(A)\setminus\mathop{\mathrm{\mathrm{colsupp}}}\nolimits(C)) \cup \{3\} = \{3,5\}$.
  • Figure :

Theorems & Definitions (96)

  • Proposition 1.1
  • Theorem 1.2: \ref{['prop:ehrhart_poly']}
  • Theorem 1.3: \ref{['thrm:ehrhart_series']}
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: \ref{['prop:harmonic-pres']}
  • Corollary 1.7
  • Theorem 1.8
  • Definition 3.1
  • Definition 3.2
  • ...and 86 more