Year: 2026
Type: preprint
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 $Δ$ is generated as an algebra by the harmonic space of $Δ$ using structural results on 2-factors of regular multigraphs. In particular, the harmonic algebra is finitely generated, giving a second proof of rationality.
Source: arXiv (Cornell University)
Authors: Libman, Nathaniel, Miller, Weston
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204528546
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