Root polytopes, flow polytopes, and order polytopes

In this paper we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points ᵢ-eⱼ \ \ i ≠ j\ ∪ \± eᵢ\ in Rⁿ, where e₁,,eₙ is the standard basis of Rⁿ. Such a polytope can be encoded by a quiver Q with vertices V ⊆ ₁,,vₙ\ ∪ \\, where each edge vⱼ→ vᵢ or → vᵢ or vᵢ→ gives rise to the point eᵢ-eⱼ or eᵢ or -eᵢ, respectively; we denote the corresponding polytope as 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 vertex) then the root polytope Root(Q) is reflexive and terminal; we moreover give a combinatorial description of the facets of Root(Q). We also show that if Q is planar, then Root(Q) is (integrally equivalent to the) polar dual of the flow polytope of the dual quiver Q^. Finally we consider the case that Q comes from the Hasse diagram of a ranked poset P, and show that Root(Q) is polar dual to (a translation of) a marked poset polytope. Additionally, we show that the face fan FQ of Root(Q) refines the normal fan N(O(P)) of the order polytope 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 FQ for any ranked poset P, in terms of a new ``canonical extension'' of P. These results have applications to mirror symmetry.