Root polytopes, flow polytopes, and order polytopes

Konstanze Rietsch, Lauren Williams

Published 2024-06-22Version 1

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 star-connected (or strongly-connected if there is no $\star$ vertex) 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 $Q^{\vee}$. Finally we consider the case that $Q$ comes from the Hasse diagram of a ranked poset $P$, and show that $\operatorname{Root}(Q)$ is polar dual to (a translation of) a marked poset polytope. Additionally, we show that the face fan $\mathcal{F}_Q$ of $\operatorname{Root}(Q)$ refines the normal fan $\mathcal{N}(\mathcal{O}(P))$ of the order polytope $\mathcal{O}(P)$, and when $P$ is graded, these fans coincide. We moreover give a combinatorial description of the Picard group of the toric variety associated to $\mathcal{F}_Q$ for any ranked poset $P$, in terms of a new ``canonical extension'' of $P$. These results have applications to mirror symmetry.