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Harmonics and graded Ehrhart theory

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arxiv 2407.06511 v3 pith:RMJQBYLO submitted 2024-07-09 math.CO math.AC

Victor Reiner , Brendon Rhoades This is my paper · ORCID
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keywords ehrhartseriesalgebraconfigurationsintroduceinverselatticemacaulay
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abstract

The Ehrhart polynomial and Ehrhart series count lattice points in integer dilations of a lattice polytope. We introduce and study a $q$-deformation of the Ehrhart series, based on the notions of harmonic spaces and Macaulay's inverse systems for coordinate rings of finite point configurations. We conjecture that this $q$-Ehrhart series is a rational function, and introduce and study a bigraded algebra whose Hilbert series matches the $q$-Ehrhart series. Defining this algebra requires a new result on Macaulay inverse systems for Minkowski sums of point configurations.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Graded Ehrhart theory and toric geometry

    math.CO 2025-08 accept novelty 8.0 of 10

    The harmonic algebra of a lattice polytope is identified with the associated graded semigroup algebra and with a blowup section ring quotient, yielding lattice triangles whose harmonic algebra is not finitely generated.

  2. Derangement permutation matrices and orbit harmonics

    math.CO 2026-07 accept novelty 6.0 of 10

    R(D_n) has generators I_n plus diagonal variables, Hilbert series sum_w q^{n-lis(Ψ(w))} over derangements, and graded Frobenius given by an alternating Kronecker formula.

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