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

Abstract

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 $n-1$ 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 $Gr_{k,k+2}$ under a map $\tilde{Z}$ induced by a matrix $Z\in \text{Mat}_{n,k+2}^{>0}$. Introduced in the context of scattering amplitudes, it is not a polytope, and has dimension $2k$. Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as was first noted by Lukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented matroid theory, total positivity, and 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 $\tilde{Z}$. Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We prove the main conjecture of (LPW): a collection of positroid polytopes is a triangulation of $\Delta_{k+1, n}$ if and only if the collection of T-dual Grasstopes is a triangulation of $\mathcal{A}_{n,k,2}(Z)$ for all $Z$. Moreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip characterization of $\mathcal{A}_{n,k,2}(Z)$, and Lukowski--Parisi--Spradlin--Volovich's conjectures on $m=2$ cluster adjacency and on generalized triangles (images of $2k$-dimensional positroid cells which map injectively into $\mathcal{A}_{n,k,2}(Z)$). Finally, we introduce new cluster structures in the amplituhedron.