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The $h^*$-polynomials of type C hypersimplices

Antoine Abram, Jose Bastidas

TL;DR

The paper extends Ehrhart theory to type $C$ hypersimplices, introducing a new statistic on signed permutations and a poset model that captures the alcove structure. It yields explicit $h^*$-polynomial formulas for half-open type-$C$ hypersimplices in terms of circular and big-ascent statistics, and connects these to type $B$ Eulerian numbers through multiple generating-function and shelling arguments. It also analyzes the $h^*$-polynomial of the fundamental parallelepiped, provides recurrences and a generating-function framework, and describes a limiting poset that converges to the lattice of strict partitions, highlighting deep connections between combinatorial statistics on signed permutations and Ehrhart theory for root-system–driven polytopes. The work thus unifies geometry, combinatorics, and asymptotic poset theory in the study of type $C$ hypersimplices and their Ehrhart invariants.

Abstract

We study the Ehrhart theory of hypersimplices of type C, as introduced by Lam and Postnikov for general crystallographic root systems. The $h^*$-polynomials of classical hypersimplices are known to relate to various Eulerian statistics on the symmetric group. In this paper, we introduce a new statistic and partial order on signed permutations, which we use to derive explicit formulas for the $h^*$-polynomials of type C hypersimplices. Additionally, we explore connections with other statistics, including flag-excedances and circular descents, flag-descents, and Coxeter descents.

The $h^*$-polynomials of type C hypersimplices

TL;DR

The paper extends Ehrhart theory to type hypersimplices, introducing a new statistic on signed permutations and a poset model that captures the alcove structure. It yields explicit -polynomial formulas for half-open type- hypersimplices in terms of circular and big-ascent statistics, and connects these to type Eulerian numbers through multiple generating-function and shelling arguments. It also analyzes the -polynomial of the fundamental parallelepiped, provides recurrences and a generating-function framework, and describes a limiting poset that converges to the lattice of strict partitions, highlighting deep connections between combinatorial statistics on signed permutations and Ehrhart theory for root-system–driven polytopes. The work thus unifies geometry, combinatorics, and asymptotic poset theory in the study of type hypersimplices and their Ehrhart invariants.

Abstract

We study the Ehrhart theory of hypersimplices of type C, as introduced by Lam and Postnikov for general crystallographic root systems. The -polynomials of classical hypersimplices are known to relate to various Eulerian statistics on the symmetric group. In this paper, we introduce a new statistic and partial order on signed permutations, which we use to derive explicit formulas for the -polynomials of type C hypersimplices. Additionally, we explore connections with other statistics, including flag-excedances and circular descents, flag-descents, and Coxeter descents.

Paper Structure

This paper contains 23 sections, 19 theorems, 130 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Signed permutations
  4. Signed permutation statistics and type B Eulerian numbers
  5. Alcoved polytopes
  6. Root systems
  7. Affine Coxeter arrangement and alcoves
  8. Alcoved polytopes and generalized hypersimplices
  9. Type C hypersimplices
  10. Proof of Theorem \ref{['t:fexc-desB']}
  11. Combinatorial Model: Poset on Signed Permutations
  12. Big ascents
  13. Poset structure
  14. Proof of Theorem \ref{['t:cdes-basc']}
  15. Type C circular descents (revisited)
  16. ...and 8 more sections

Key Result

Theorem 1

For $n \geq 1$ and $k \geq 1$, the $h^*$-polynomial of the half-open hypersimplex $\Delta'_{C_n,k}$ is

Figures (4)

  • Figure 1: Left: Positive roots of type $C_2$. Right: The affine arrangement $\mathcal{H}_{\Phi_{C_2}}$ with the fundamental parallelepiped $\Pi_{C_2}$ subdivided into the three hypersimplices $\Delta_{C_2,1}$ (red, bottom), $\Delta_{C_2,2}$ (blue, center), and $\Delta_{C_2,3}$ (orange, top). Only thick or dashed hyperplanes are labeled.
  • Figure 2: The poset $(X_n , \leq)$ for $n=2$ (left) and $n=3$ (right) with the big ascent positions $i \in \{\overline{1}\} \cup [n]$ underlined with a little wedge "$\widehat{\space}$". Elements $w \in X_n$ are grouped by the value of $\mathop{\mathrm{cdes}}\nolimits(w^{-1})$.
  • Figure 3: The first values of $\psi_{n,k}$ computed using the recurrence in \ref{['eq:recurrence']}. For instance, $\psi_{5,2} = (2 \cdot 5 - 2 \cdot 2 + 1) \cdot 49 + (2 \cdot 2 + 1) \cdot 115 = 918$. Each number is the weighted sum of the values directly north-east and north-west of it, and the weights are constant along the diagonals. See the last row of \ref{['tab:h-half-open']} for more values of $\psi_{n,k}$.
  • Figure 4: Subposets $(Y_n , \leq)$ for $n = 2,3$ (left), and their embedding in the lattice of strict partitions (right), represented using shifted diagrams (in French notation).

Theorems & Definitions (53)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Example 2: Type A
  • Example 3: Type C
  • Definition 4: Lam--Postnikov
  • Definition 5: Lam--Postnikov
  • ...and 43 more