Volume, facets and dual polytopes of twinned chain polytopes

TL;DR

This paper investigates the combinatorial properties of twinned chain polytopes, providing formulas for their volume, describing their facets, and characterizing their dual polytopes, thus advancing understanding of their geometric structure.

Contribution

It introduces explicit formulas for volume, facet-supporting hyperplanes, and vertex representations of duals of twinned chain polytopes based on underlying posets.

Findings

Volume formula in terms of posets

Facet-supporting hyperplanes characterized

Vertex representations of dual polytopes provided

Abstract

Let $(P,\leq_P)$ and $(Q,\leq_Q)$ be finite partially ordered sets with $|P|=|Q|=d$, and $\mathcal{C}(P) \subset \mathbb{R}^d$ and $\mathcal{C}(Q) \subset \mathbb{R}^d$ their chain polytopes. The twinned chain polytope of $P$ and $Q$ is the lattice polytope $\Gamma(\mathcal{C}(P),\mathcal{C}(Q)) \subset \mathbb{R}^d$ which is the convex hull of $\mathcal{C}(P) \cup (-\mathcal{C}(Q))$. It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain polytopes in terms of the underlying partially ordered sets. Second, we will identify the facet-supporting hyperplanes of twinned chain polytopes in terms of the underlying partially ordered sets. Finally, we will provide the vertex representations of the dual polytopes of twinned chain polytopes.