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2-levelness of Marked Poset Polytopes and the Ehrhart polynomial

Jan Stricker

TL;DR

The paper investigates 2-levelity for marked poset polytopes and derives an explicit Ehrhart polynomial for marked order polytopes. It establishes that marked order and marked chain polytopes are 2-level precisely when they are affinely isomorphic to ordinary order or chain polytopes, respectively, and extends a similar characterization to marked chain-order polytopes. A central result is an exact Ehrhart formula for $\\mathcal{O}(P,\\lambda)$ expressed via linear extensions and marked chains, which, by Ehrhart equivalence, also yields the Ehrhart polynomials for the related marked chain and chain-order polytopes. The work connects polyhedral combinatorics with poset structure and has implications for representations tied to PBW-degenerations and related geometric objects, offering concrete counting formulas for lattice points in these families of polytopes. The results provide a unified framework for understanding 2-levelness across several marked poset polytope families and supply practical tools for computing their lattice-point enumerators.

Abstract

It is already known that order polytopes and chain polytopes are always 2-level polytopes. In general, this is not true for marked order and marked chain polytopes. We study the geometry of marked order polytopes, marked chain polytopes, and marked chain-order polytopes, providing a comprehensive characterisation of 2-levelness for these polytopes. Furthermore, we present an exact formula for the Ehrhart polynomial of marked order polytopes. Because of their connection to marked chain and marked chain-order polytopes, this polynomial is also the Ehrhart polynomial of these polytopes.

2-levelness of Marked Poset Polytopes and the Ehrhart polynomial

TL;DR

The paper investigates 2-levelity for marked poset polytopes and derives an explicit Ehrhart polynomial for marked order polytopes. It establishes that marked order and marked chain polytopes are 2-level precisely when they are affinely isomorphic to ordinary order or chain polytopes, respectively, and extends a similar characterization to marked chain-order polytopes. A central result is an exact Ehrhart formula for expressed via linear extensions and marked chains, which, by Ehrhart equivalence, also yields the Ehrhart polynomials for the related marked chain and chain-order polytopes. The work connects polyhedral combinatorics with poset structure and has implications for representations tied to PBW-degenerations and related geometric objects, offering concrete counting formulas for lattice points in these families of polytopes. The results provide a unified framework for understanding 2-levelness across several marked poset polytope families and supply practical tools for computing their lattice-point enumerators.

Abstract

It is already known that order polytopes and chain polytopes are always 2-level polytopes. In general, this is not true for marked order and marked chain polytopes. We study the geometry of marked order polytopes, marked chain polytopes, and marked chain-order polytopes, providing a comprehensive characterisation of 2-levelness for these polytopes. Furthermore, we present an exact formula for the Ehrhart polynomial of marked order polytopes. Because of their connection to marked chain and marked chain-order polytopes, this polynomial is also the Ehrhart polynomial of these polytopes.
Paper Structure (6 sections, 16 theorems, 40 equations, 3 figures)

This paper contains 6 sections, 16 theorems, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of relevant Theorems and Definitions
  3. 2-level Marked Order Polytopes
  4. 2-level Marked Chain Polytopes
  5. 2-level Marked Chain-Order Polytope
  6. Ehrhart polynomial of marked order polytopes

Key Result

Proposition 2.2

Let $Q \subseteq \mathbb{R}^d$ be a polytope. Then $Q$ is a 2-level polytope, if one of the following is satisfied: (i) For each hyperplanes $H$, which define a facet $F$, (ii) Let $\{x \in \mathbb{R}^d\,:\,a_i^tx\leq b_i\, ,\,i\in [m]\}$ be a finite irredundant description for the polytope $Q$ with $a_i \in \mathbb{R}^d$ and $b_i\in \mathbb{R}$. For all $i \in [m]$ there exists $m_i \in \mathbb{

Figures (3)

  • Figure 1: The marked poset $(P,\lambda)$ from Example \ref{['bsplem3-1']} and the corresponding marked chain polytope $\mathcal{C}(P,\lambda)$.
  • Figure 2: The poset $P'$ from Example \ref{['bsplem3-1']} and the corresponding chain polytope $\mathcal{C}_{P'}$.
  • Figure 3: Examples of the marked posets from Example \ref{['bsplem5-1']}.

Theorems & Definitions (42)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9
  • ...and 32 more