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Simplex inequalities of order and chain polytopes of recursively defined posets

Ragnar Freij-Hollanti, Teemu Lundström

TL;DR

This paper addresses the problem of comparing simplex faces between the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ of a finite poset $P$, within the Hibi–Li framework of $f$-vector inequalities. By developing polynomial tools $S_P(x)$ and its origin-containing variants, and analyzing subdirect sums (ordinal sums) and Cartesian products, the authors establish that for posets $P$ built from $X$-free seeds via disjoint unions and ordinal sums (the class $\mathcal{F}$), the inequality $s_k(\mathcal{O}(P)) \le s_k(\mathcal{C}(P))$ holds for all $k \ge 0$. Equality occurs precisely when $P$ is $X$-free, linking to unimodular equivalence and extending Mori's result from triangles to all dimensions within $\mathcal{F}$. The work connects to the Hibi–Li conjecture and broad $f$-vector inequalities, and highlights open questions about the maximal simplex dimension and potential further characterizations of equality cases. Overall, the paper significantly broadens the understanding of simplex-face distributions between poset polytopes and provides a robust framework for comparing these combinatorial-geometric invariants.

Abstract

In this paper, we study the simplex faces of the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ of a finite poset $P$. We show that, if $P$ can be recursively constructed from $\mathbf{X}$-free posets using disjoint unions and ordinal sums, then $\mathcal{C}(P)$ has at least as many $k$-dimensional simplex faces as $\mathcal{O}(P)$ does, for each dimension $k$. This generalizes a previous result of Mori, both in terms of the dimensions of the simplices and in terms of the class of posets considered.

Simplex inequalities of order and chain polytopes of recursively defined posets

TL;DR

This paper addresses the problem of comparing simplex faces between the order polytope and the chain polytope of a finite poset , within the Hibi–Li framework of -vector inequalities. By developing polynomial tools and its origin-containing variants, and analyzing subdirect sums (ordinal sums) and Cartesian products, the authors establish that for posets built from -free seeds via disjoint unions and ordinal sums (the class ), the inequality holds for all . Equality occurs precisely when is -free, linking to unimodular equivalence and extending Mori's result from triangles to all dimensions within . The work connects to the Hibi–Li conjecture and broad -vector inequalities, and highlights open questions about the maximal simplex dimension and potential further characterizations of equality cases. Overall, the paper significantly broadens the understanding of simplex-face distributions between poset polytopes and provides a robust framework for comparing these combinatorial-geometric invariants.

Abstract

In this paper, we study the simplex faces of the order polytope and the chain polytope of a finite poset . We show that, if can be recursively constructed from -free posets using disjoint unions and ordinal sums, then has at least as many -dimensional simplex faces as does, for each dimension . This generalizes a previous result of Mori, both in terms of the dimensions of the simplices and in terms of the class of posets considered.

Paper Structure

This paper contains 4 sections, 17 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Counting simplex faces
  4. Main result and conclusion

Key Result

Proposition 2.1

Let $\mathcal{P}$ and $\mathcal{Q}$ be polytopes both containing the origin as a vertex. The faces of $\mathcal{P} \vee \mathcal{Q}$ are

Figures (1)

  • Figure 1: The smallest poset $\mathbf{X}$ for which $\mathcal{O}(\mathbf{X})$ and $\mathcal{C}(\mathbf{X})$ are not unimodularly equivalent.

Theorems & Definitions (28)

  • Proposition 2.1: f-vector_inequalities
  • Proposition 2.2: f-vector_inequalities
  • Theorem 2.3: cutting
  • Theorem 2.4: mori_simplex_faces
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • ...and 18 more