Combinatorics Seminar

I'll discuss a surprising parallel between certain decompositions of the amplituhedron, a non-polytopal subset of a Grassmannian, and the hypersimplex, a polytope in R^n. The amplituhedron was introduced by physicists Arkani-Hamed and Trnka to better understand scattering amplitudes in N=4 super Yang-Mills theory. In particular, each "fine positroidal" decomposition of the amplituhedron conjecturally gives you a way to compute a scattering amplitude. The hypersimplex is a classical object in algebraic combinatorics; its decompositions correspond to tropical linear spaces (Speyer) and are parametrized by the Dressian. Despite the dissimilarities of the hypersimplex and the m=2 amplituhedron, Lukowski--Parisi--Williams conjectured a straightforward bijection between their fine positroidal decompositions. I'll discuss joint work with Matteo Parisi and Lauren Williams, in which we prove this bijection. Along the way, we prove an intrinsic description of the m=2 amplituhedron, originally conjectured by Arkani-Hamed--Thomas--Trnka; give a decomposition of the m=2 amplituhedron into chambers enumerated by the Eulerian numbers, in direct analogy with a triangulation of the hypersimplex; and find new cluster varieties in the Grassmannian. Speaker(s): Melissa Sherman-Bennet (University of Michigan)