Rationality of Poincar\'e Series for a Family of Lattice Simplices

TL;DR

This paper proves the rationality of multigraded Poincaré series for a family of Gorenstein semigroup algebras linked to reflexive lattice simplices, revealing complex interactions between generating functions and algebraic structures.

Contribution

It introduces explicit minimal free resolutions for these algebras, showing their non-Koszul nature and the non-triviality of Poincaré series rationality.

Findings

Poincaré series are rational for the studied algebras.

The algebras are not Koszul.

Resolutions reflect recursive structure in the denominator.

Abstract

We investigate multi-graded Gorenstein semigroup algebras associated with an infinite family of reflexive lattice simplices. For each of these algebras, we prove that their multigraded Poincar\'e series is rational. Our method of proof is to produce for each algebra an explicit minimal free resolution of the ground field, in which the resolution reflects the recursive structure encoded in the denominator of the finely-graded Poincar\'e series. Using this resolution, we show that these algebras are not Koszul, and therefore rationality is non-trivial. Our results demonstrate how interactions between multivariate and univariate rational generating functions can create subtle complications when attempting to use rational Poincar\'e series to inform the construction of minimal resolutions.