Vertex Posets, Monotone Path Polytopes, and Chow Polynomials

TL;DR

This paper explores duality in partitions of convex polytopes induced by linear functionals, linking poset invariants with monotone path polytopes and Chow polynomials.

Contribution

It establishes a duality criterion for stratifications and connects poset invariants with the combinatorics of monotone path polytopes.

Findings

Positive and negative partitions are dual stratifications under certain conditions.

Chow polynomial of a vertex poset equals the h-polynomial of a monotone path polytope.

Results connect poset invariants with polytope combinatorics.

Abstract

Let $P\subset\mathbb R^n$ be a convex polytope and let $\ell$ be a linear functional which is nonconstant on every edge of $P$. The induced acyclic orientation determines positive and negative Bia{\l}ynicki-Birula type partitions of $P$ into unions of relative interiors of faces. Our first result establishes a duality: the positive partition is a stratification if and only if the negative one is a stratification. Our second result connects poset invariants with monotone path polytopes. Assuming the induced vertex relation admits the structure of a graded poset, we prove that the Chow polynomial of the resulting vertex poset agrees with the $h$-polynomial of a (dual) monotone path polytope.