Gorenstein simplices and the associated finite abelian groups

It is known that a lattice simplex of dimension d corresponds a finite abelian subgroup of (R∕Z)d+1. Conversely, given a finite abelian subgroup of (R∕Z)d+1 such that the sum of all entries of each element is an integer, we can obtain a lattice simplex of dimension d. In this paper, we discuss a characterization of Gorenstein simplices in terms of the associated finite abelian groups. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equals p,p2 and pq, where p and q are prime numbers with p≠q. Moreover, we compute the volume of the associated dual reflexive simplices of the Gorenstein simplices.