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 Goren stein simplices in terms of the associated finite abelian groups. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equals p, p(2) and pq, where p and q are prime numbers with p not equal q. Moreover, we compute the volume of the associated dual reflexive simplices of the Gorenstein simplices. (C) 2017 Elsevier Ltd. All rights reserved.