Séminaire Lotharingien de Combinatoire, 95B.41 (2026), 12 pp.
Colin Crowley and Ethan Partida
Graded Ehrhart Theory of 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. Our work answers, in the special case of unimodular zonotopes, two conjectures of Reiner and Rhoades (2024).
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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