The Hypersimplex

Abstract

In 1987, the foundational work of Gelfand-Goresky-MacPherson-Serganova initiated the study of the Grassmannian and torus orbits in the Grassmannian via the moment map and matroid polytopes, which arise as moment map images of (closures of) torus orbits. The moment map image of the (positive) Grassmannian ${{\,\textrm{Gr}\,}}_{k+1,n}$ is the $(n-1)$-dimensional hypersimplex $\Delta _{k+1,n} \subseteq \mathbb {R}^n$, the convex hull of the indicator vectors $e_I\in \mathbb {R}^n$ where $I \in {[n] \atopwithdelims ()k+1}$. In this chapter we introduce and present new results about the hypersimplex and positroid polytopes—(closures of) images of positroid cells of ${{\,\textrm{Gr}\,}}^{\ge 0}_{k+1,n}$ under the moment map. We give a full characterization of positroid polytopes which are images of positroid cells where the moment map is injective. In particular, the full-dimensional ones— we call positroid tiles—are in bijection with plabic trees. We then consider the problem of finding positroid tilings—collections of positroid tiles whose interiors are pair-wise disjoint and cover the hypersimplex. We show that the positive tropical Grassmannain ${{\,\textrm{Trop}\,}}^+{{\,\textrm{Gr}\,}}_{k+1,n}$ is the secondary fan for regular subdivisions of $\Delta _{k+1,n}$ into positroid polytopes. In particular, maximal cones of ${{\,\textrm{Trop}\,}}^+{{\,\textrm{Gr}\,}}_{k+1,n}$ are in bijection with regular positroid tilings of $\Delta _{k+1,n}$.