Classification and counting of Gorenstein simplices with $h^*$-polynomial $1+t^k+\cdots+t^{(v-1)k}$

Abstract

Hibi, Yoshida, and the author classified Gorenstein simplices which are not lattice pyramids and whose \(h^*\)-polynomials are of the form \(1+t^k+t^{2k}+\cdots+t^{(v-1)k}\) when \(v\) is a prime number or the product of two prime numbers. They also conjectured that, for general \(v\), the number of unimodular equivalence classes of such simplices depends only on the divisor lattice of \(v\). This paper proves the conjecture by giving a constructive classification of Gorenstein simplices whose \(h^*\)-polynomials are of this form. More precisely, their unimodular equivalence classes are shown to be parametrized by strict divisor chains in the divisor lattice of \(v\) together with certain recursive combinatorial data. As a consequence, an explicit formula for the number of equivalence classes in terms of the divisor lattice of \(v\) is obtained.