Abstract

I’ll discuss a curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian (under the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. In joint work with Matteo Parisi and Lauren Williams, we give a bijection between decompositions of the amplituhedron into “postroid tiles” and decompositions of the hypersimplex into positroid polytopes (originally conjectured by Lukowski—Parisi—Williams). Along the way, we find a cluster connection: the positroid tiles are the positive parts of new cluster varieties in Gr(k, k+2). We also prove a sign-flip description of the amplituhedron conjectured by Arkani-Hamed—Thomas—Trnka and introduce a new, finer decomposition of the amplituhedron into Eulerian-number-many chambers.