Triangular faces of the order and chain polytope of a maximal ranked poset

TL;DR

The paper studies the relationship between the face structures of the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ of a finite poset $P$, focusing specifically on triangular $2$-faces. It proves that for maximal ranked posets, the number of triangular $2$-faces satisfies $|\Delta_{\mathcal{O}(P)}| \le |\Delta_{\mathcal{C}(P)}|$, with equality exactly when $P$ contains no $X$-poset, linking this to unimodular equivalence. The method centers on classifying triangular $2$-faces via sets $\Delta_{\mathcal{O}(P)}$, $\Delta_{\mathcal{C}(P)}$, and their edge-relations $E^{*}_{\mathcal{O}(P)}$, $E^{*}_{\mathcal{C}(P)}$, and constructing an injection $\varphi$ from $\Delta^{*}_{\mathcal{O}(P)}$ to $\Delta^{*}_{\mathcal{C}(P)}$ to compare counts; presence of an $X$-poset yields a strict inequality and a combinatorial surplus formula. Overall, the results advance Hibi–Li style conjectures within the class of maximal ranked posets and clarify how $X$-posets drive asymmetry between $\mathcal{O}(P)$ and $\mathcal{C}(P)$.

Abstract

Let $\mathscr{O}(P)$ and $\mathscr{C}(P)$ denote the order polytope and chain polytope, respectively, associated with a finite poset $P$. We prove the following result: if $P$ is a maximal ranked poset, then the number of triangular $2$-faces of $\mathscr{O}(P)$ is less than or equal to that of $\mathscr{C}(P)$, with equality holding if and only if $P$ does not contain an $X$-poset as a subposet.

Figures (1)

• Figure 1: $X$-poset

Theorems & Definitions (16)

• Conjecture 1.1: HL2
• Lemma 2.1: HL1
• Lemma 2.2: M
• Remark 2.3
• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.3
• proof
• Lemma 3.4