The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers

Matteo Parisi, Melissa Sherman-Bennett, Lauren Williams

Published 2021-04-16Version 1

The hypersimplex $\Delta_{k+1,n}$ is the image of the positive Grassmannian $Gr^{\geq 0}_{k+1,n}$ under the moment map. It is a polytope of dimension $n-1$ which lies in $\mathbb{R}^n$. Meanwhile, the amplituhedron $\mathcal{A}_{n,k,2}(Z)$ is the projection of the positive Grassmannian $Gr^{\geq 0}_{k,n}$ into the Grassmannian $Gr_{k,k+2}$ under the amplituhedron map $\tilde{Z}$. It is not a polytope, and has full dimension $2k$ inside $Gr_{k,k+2}$. Nevertheless, as was first discovered by Lukowski-Parisi-Williams [LPW], these two objects appear to be closely related via T-duality. In this paper we use the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes - images of positroid cells of $Gr^{\geq 0}_{k+1,n}$ under the moment map - translate into sign conditions characterizing the T-dual Grasstopes - images of positroid cells of $Gr^{\geq 0}_{k,n}$ under the amplituhedron map. Moreover, we subdivide the amplituhedron into chambers enumerated by the Eulerian numbers, just as the hypersimplex can be subdivided into simplices enumerated by the Eulerian numbers. We use this property to prove one direction of the conjecture of [LPW]: whenever a collection of positroid polytopes gives a triangulation of the hypersimplex, the T-dual Grasstopes give a triangulation of the amplituhedron. Along the way, we prove: Arkani-Hamed-Thomas-Trnka's conjecture that $\mathcal{A}_{n,k,2}(Z)$ can be characterized using sign conditions, and Lukowski-Parisi-Spradlin-Volovich's conjectures about characterizing generalized triangles (the $2k$-dimensional positroid cells which map injectively into the amplituhedron $\mathcal{A}_{n,k,2}(Z)$), and $m=2$ cluster adjacency. Finally, we discuss new cluster structures in the amplituhedron.