We give two new constructions of the harmonic algebra of a lattice polytope P P , a bigraded algebra whose character is the q q -Ehrhart series of P P defined by Reiner and Rhoades Harmonics and graded Ehrhart theory, arXiv: 2407.06511 , 2024. First, we show that the harmonic algebra is the associated graded algebra of the semigroup algebra of P P with respect to a certain natural filtration, clarifying it’s relationship with the more classical semigroup algebra. We then give a geometric interpretation of the harmonic algebra as a quotient of the ring of global sections of a certain family of line bundles on the blowup of the toric variety associated to P P at a generic point. Using this connection to toric geometry we resolve one the main conjectures of Reiner and Rhoades by showing that the harmonic algebra is not finitely generated in general.
Ian Cavey (Thu,) studied this question.