Root polytopes, flow polytopes, and order polytopes

Abstract

In this paper we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb{R}^n$, where $e_1,\dots,e_n$ is the standard basis of $\mathbb{R}^n$. Such a polytope can be encoded by a quiver $Q$ with vertices $V \subseteq \{v_1,\dots,v_n\} \cup \{\star\}$, where each edge $v_j\to v_i$ or $\star \to v_i$ or $v_i\to \star$ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\operatorname{Root}(Q)$. These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver $Q$ is strongly-connected then the root polytope $\operatorname{Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\operatorname{Root}(Q)$. We also show that if $Q$ is planar, then $\operatorname{Root}(Q)$ is (integrally equivalent to the) polar dual of the flow polytope of the dual quiver. Finally we consider the case that $Q$ comes from a ranked poset $P$, and show that $\operatorname{Root}(Q)$ is polar dual to (a translation of) a marked poset polytope. We then study the toric variety $Y(\mathcal{F}_Q)$ associated to the face fan $\mathcal{F}_Q$ of $\operatorname{Root}(Q)$. If $Q$ comes from a ranked poset $P$ we give a combinatorial description of the Picard group of $Y(\mathcal{F}_Q)$, and we show that $Y(\mathcal{F}_Q)$ is a small partial desingularisation of the Hibi toric variety $Y_{\mathcal{O}(P)}$ of the order polytope $\mathcal{O}(P)$. We show that $Y(\mathcal{F}_Q)$ has a small crepant toric resolution of singularities $Y(\widehat{\mathcal{F}}_Q)$, and as a consequence that the Hibi toric variety $Y_{\mathcal{O}(P)}$ has a small resolution of singularities for any ranked poset $P$. These results have applications to mirror symmetry.