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We investigate multi-graded Gorenstein semigroup algebras associated with aninfinite family of reflexive lattice simplices. For each of these algebras, weprove that their multigraded Poincar\'e series is rational. Our method of proofis to produce for each algebra an explicit minimal free resolution of theground field, in which the resolution reflects the recursive structure encodedin the denominator of the finely-graded Poincar\'e series. Using thisresolution, we show that these algebras are not Koszul, and thereforerationality is non-trivial. Our results demonstrate how interactions betweenmultivariate and univariate rational generating functions can create subtlecomplications when attempting to use rational Poincar\'e series to inform theconstruction of minimal resolutions.

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