Graded Ehrhart theory of unimodular zonotopes

There is a well-studied connection between Hilbert series of toric varieties (algebraic geometry) and Ehrhart theory (combinatorics), which gives beautiful geometric meaning to concrete identities of generating functions associated to polytopes. Recently, Reiner and Rhoades stated conjectural q-analogues of the main theorems in Ehrhart theory, and Cavey provided a connection to the geometry of blown-up toric varieties. While Reiner and Rhoades's conjectures are false in general, we prove that they are true for the class of multiplicity-free subtoric varieties of a product of projective lines, which correspond to unimodular zonotopes. We prove our results using an alternate geometric model, known as an arrangement Schubert variety, which plays a central role in recent developments in matroid theory. Based on joint work with Ethan Partida.

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