Two-Dimensional Faces of Order and Chain Polytopes

To a finite poset $P$ with $n$ elements, Stanley introduced two $n$-dimensional polytopes, the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$. These polytopes are known to have the same volumes and the same numbers of vertices and edges, while $\mathcal{C}(P)$ is known to have strictly more facets than $\mathcal{O}(P)$, except in the cases where the polytopes are unimodularly equivalent. In this paper, we study the structure of the two-dimensional faces of the two polytopes. We give combinatorial parametrizations of the $2$-dimensional faces of both polytopes. In the case of the order polytope, these parametrizations are adaptions of descriptions of faces by Geissinger and Stanley, while in the case of the chain polytope, the characterizations themselves are novel. To parametrize the triangle faces in particular, we introduce the notions of biconnected filters and biconnected antichains in a poset, of possible independent interest. Using these parametrizations, we show that for any $P$, $\mathcal{C}(P)$ has equally many square faces, and at least as many triangular faces, as $\mathcal{O}(P)$ does. Moreover, the inequality is shown to be strict except when $\mathcal{O}(P)$ and $\mathcal{C}(P)$ are unimodularly equivalent. This proves the case $i=2$ of a conjecture by Hibi and Li.