Root polytopes, flow polytopes, and order polytopes

TL;DR

This paper explores a class of polytopes derived from quivers, revealing their geometric properties, dualities, and associated toric varieties, with implications for mirror symmetry and singularity resolutions.

Contribution

It introduces a comprehensive study of root polytopes from quivers, detailing their geometric features, dualities, and connections to poset polytopes and toric varieties, including new resolution results.

Findings

Root polytopes are reflexive and terminal for strongly-connected quivers.

Planar quivers' root polytopes are dual to flow polytopes of dual quivers.

The associated toric varieties admit small crepant resolutions.

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.