# Gorenstein simplices and the associated finite abelian groups

## TL;DR

This paper characterizes Gorenstein simplices via associated finite abelian groups, providing classifications for specific volume cases and computing volumes of their duals, advancing understanding in lattice polytope theory.

## Contribution

It offers a complete characterization of Gorenstein simplices with certain volumes and explores their duals, linking lattice geometry with finite abelian group structures.

## Key findings

- Characterization of Gorenstein simplices with volume p, p^2, and pq
- Complete classification of these simplices based on associated groups
- Volume calculations for dual Gorenstein simplices

## Abstract

It is known that a lattice simplex of dimension $d$ corresponds a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$. Conversely, given a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$ 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 $p,p^2$ and $pq$, where $p$ and $q$ are prime numbers with $p \neq q$. Moreover, we compute the volume of the dual simplices of Gorenstein simplices.