Faces of maximal chain polytopes

TL;DR

This paper investigates the structure of maximal chain polytopes associated with finite posets, identifying conditions under which certain convex hulls form faces and applying these results to specific examples like product posets.

Contribution

It characterizes when convex hulls of maximal chains are faces of the polytope and computes examples such as the dimension of product posets.

Findings

Convex hulls of maximal chains are not faces if they have an incomplete guided crown structure.

The paper provides explicit calculations for the dimension of maximal chain polytopes of product posets.

Identifies structural conditions affecting face formation in maximal chain polytopes.

Abstract

The maximal chain polytope $\mathscr{M}(P)$ is associated with a finite poset $P$. For a set of maximal chains $\mathcal{C}$, it is shown that the convex hull of all the points corresponding to elements of $\mathcal{C}$ is not a face of $\mathscr{M}(P)$ if and only if $\mathcal{C}$ has an incomplete guided crown structure. Using this result, several examples, including the dimension of $\mathscr{M}(\boldsymbol{m} \times \boldsymbol{n})$, are calculated.