Detecting the Integer Decomposition Property and Ehrhart Unimodality in Reflexive Simplices

TL;DR

This paper investigates the relationship between the integer decomposition property and Ehrhart unimodality in reflexive simplices, providing new proofs and extending known results to broader classes of lattice simplices.

Contribution

It proves the conjecture for a natural generalization of Payne's counterexamples and explores unimodality in a broader class of reflexive lattice simplices.

Findings

Confirmed the conjecture for a generalized class of Payne's examples

Extended the analysis to new families of lattice simplices

Presented open problems for future research

Abstract

A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) $h^\ast$-polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal $h^\ast$-polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Musta{\c{t}}{\v{a}} and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous examples in support of the conjecture that IDP reflexives are $h^\ast$-unimodal, its validity has not yet been considered for families of reflexive lattice simplices that closely generalize Payne's counterexamples. The main purpose of this work is to prove that the former conjecture does indeed hold for a natural generalization of Payne's examples. The second purpose of this work is to extend this investigation to a broader class of lattice simplices, for which we present new results and open problems.