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arXiv · 2608.27438

Graded Ehrhart theory for hypersimplices

Nathaniel Libman·Weston Miller
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Abstract

We prove that the $q$-Ehrhart series of a hyperplane slice of a cube is a rational function with an explicit denominator that satisfies $q$-reciprocity, confirming a conjecture of Reiner and Rhoades for these polytopes. To do this, we find a generating set for the orbit harmonics ideal, which also yields the Hilbert series and graded Frobenius characteristic of the associated quotient. We further show that the harmonic algebra of a hypersimplex $\Delta$ is generated as an algebra by the harmonic space of $\Delta$ using structural results on 2-factors of regular multigraphs. In particular, the harmonic algebra is finitely generated, giving a second proof of rationality.

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Nathaniel Libman, Weston Miller. 2026-08-27. Graded Ehrhart theory for hypersimplices. https://arxiv.org/abs/2608.27438

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