Do alcoved lattice polytopes have unimodal h*-vector?

TL;DR

This paper proves that h*-vectors of certain alcoved polytopes are unimodal under specific geometric conditions, providing new insights into their combinatorial structure and connecting to previous research.

Contribution

It establishes unimodality of h*-vectors for alcoved polytopes with interior lattice points and facet distance conditions, and determines the maximal lattice distance for such polytopes.

Findings

h*-vectors are unimodal if polytopes contain interior lattice points and facets are close to interior points

maximal lattice distance for alcoved polytopes is dimension minus one

serves as a guide to previous work on unimodality of h*-vectors

Abstract

We show that h*-vectors of alcoved polytopes P in R^n (of Lie type A) are unimodal if they contain interior lattice points and their facets have lattice distance 1 to the set of interior lattice points. The maximal possible such distance for general alcoved polytopes is shown to be dim(P)-1. A secondary purpose of the paper is to serve as a guide to previous work surrounding unimodality of h*-vectors of alcoved polytopes and related questions.