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The Ehrhart series of alcoved polytopes

Elisabeth Bullock, Yuhan Jiang

TL;DR

This paper develops a shelling-based method to compute the Ehrhart series of alcoved polytopes by decomposing them into disjoint half-open alcoves, with the resulting series given by $\mathrm{Ehr}(P,z)=\frac{\sum_{w\in V} z^{\mathrm{wt}(w)}}{\prod_{i=0}^n (1-z^{\ell_i})}$, where edge weights depend only on the root system. The approach leverages the Coxeter complex and a breadth-first shelling order to reduce counting lattice points to a sum over alcoves, yielding additive Ehrhart data. For the hypersimplex $\Delta_{2,n}$, the authors connect this shelling framework to decorated ordered set partitions, providing a bijective explanation for the $h^*$-polynomial coefficients and linking to known combinatorial descriptions. Overall, the work unifies Ehrhart theory for alcoved polytopes with Coxeter-theoretic shellings and hypersimplex combinatorics, offering concrete computational tools and new combinatorial insights.

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 $Δ_{2,n}$.

The Ehrhart series of alcoved polytopes

TL;DR

This paper develops a shelling-based method to compute the Ehrhart series of alcoved polytopes by decomposing them into disjoint half-open alcoves, with the resulting series given by , where edge weights depend only on the root system. The approach leverages the Coxeter complex and a breadth-first shelling order to reduce counting lattice points to a sum over alcoves, yielding additive Ehrhart data. For the hypersimplex , the authors connect this shelling framework to decorated ordered set partitions, providing a bijective explanation for the -polynomial coefficients and linking to known combinatorial descriptions. Overall, the work unifies Ehrhart theory for alcoved polytopes with Coxeter-theoretic shellings and hypersimplex combinatorics, offering concrete computational tools and new combinatorial insights.

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 .

Paper Structure

This paper contains 11 sections, 17 theorems, 22 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgements
  4. Preliminaries
  5. Root systems
  6. Alcoved polytopes
  7. Coxeter Groups
  8. Ehrhart series of half-open rational simplices
  9. A shelling order
  10. Proof of the main theorem
  11. The hypersimplex $\Delta_{2,n}$

Key Result

Theorem 1

Fix an irreducible crystallographic root system $\Phi\subset V$, where $\dim(V)=n$. Let $P$ be an alcoved polytope and let $\Gamma_P=(V,E)$ be the dual graph to the alcove triangulation of $P$. Pick some $v_0\in V$ and orient the edges of $\Gamma_P$ so that for all $\{u,w\}\in E$, $u\to w$ if and on where $\mathop{\mathrm{wt}}\nolimits(w) = \sum_{u \to w} \mathop{\mathrm{wt}}\nolimits((u,w))$ is t

Figures (7)

  • Figure 1: On the left, we have the fundamental alcove of type $B_2$. On the right, we have an alcoved polytope of type $B_2$ and the graph of its alcoved triangulation. The Ehrhart series of the square is $\frac{1+z+z^2}{(1-z)^2 (1-z^2)}$.
  • Figure 2: On the left, we have the fundamental alcove of type $G_2$. On the right, we have an alcoved polytope of type $G_2$. The Ehrhart series of the trapezoid is $\frac{1+z^2+z^3}{(1-z)(1-z^2)(1-z^3)}$.
  • Figure 3: The generalized hypersimplex for $\Phi = B_3$ and $k = 2$. The Ehrhart series of $\Delta^{B_3}_2$ is $\mathop{\mathrm{Ehr}}\nolimits(\Delta^{B_3}_2,z) = \frac{1+z+3z^2+z^3}{(1-z)^2 (1-z^2)^2}$
  • Figure 4: The winding vector of $((1,5,6)_2,(3,7,8,11)_3,(4,10,12)_1,(2,9)_1)$ is (6,3,3,2,0,2,0,4,6,4,3,2) and the winding number is 35/7=5. The $i$-th entry of the winding vector is the circular distance between $i$ and $i+1$ in clockwise direction. One can walk from 1 to 2 to … $n$ back to 1 in clockwise direction, and the winding number is the number of times that one walks around the circle.
  • Figure 5: $\Delta_{2,5}$ with $A_0$ at the center. The arrows indicate cover relations in $\mathcal{P}_{A_0}$, pointing in increasing directions. The orange arrows are facets representing the cover relations of the alcove labeled by $((13)_1(245)_1)$.
  • ...and 2 more figures

Theorems & Definitions (53)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 3.1: Theorem 1.3, stanley80
  • ...and 43 more