The Ehrhart series of alcoved polytopes

TL;DR

This paper introduces a method to compute the Ehrhart series of alcoved polytopes using a specific shelling order, linking combinatorial structures to geometric properties.

Contribution

It presents a novel approach for calculating Ehrhart series of alcoved polytopes through shelling orders and connects this to decorated ordered set partitions.

Findings

Method for computing Ehrhart series via shelling order

Connection between shelling order and decorated set partitions

Applicable to hypersimplex $ riangle_{2,n}$

Abstract

Alcoved polytopes are convex polytopes, which are the closure of a union of alcoves in an affine Coxeter arrangement. They are rational polytopes and, therefore, have Ehrhart quasipolynomials. Here we describe a method for computing the generating function of the Ehrhart quasipolynomial, or Ehrhart series, of any alcoved polytope via a particular shelling order of its alcoves. We also show a connection between Early's decorated ordered set partitions and this shelling order for the hypersimplex $\Delta_{2,n}$.