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.