2016/08/31 by Steven N. Karp, Steven N Karp, Lauren K. Williams +1
Materials Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Basis (linear algebra) #Bounded function #Computation #Geometric and Algebraic Topology #Grassmannian #Hyperplane #Order (exchange) #Point (geometry) #Quasicrystal Structures and Properties #Sign (mathematics) #hep-th #math.CO
paper · pdf · doi:10.1093/imrn/rnx140
published as Int. Math. Res. Not. IMRN (2019), no. 5, 1401-1462 · 50 pages. v2: Final version
openalex created_date 2016/09/16 · openalex publication_date 2017/06/14 · arxiv created 2019/04/10 · arxiv updated 2021/06/10 · openalex updated_date 2026/08/06
The (tree) amplituhedron|An,k,m| is the image in the Grassmannian |Grk,k+m| of the totally nonnegative part of |Grk,n|, under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in |N=4| supersymmetric Yang–Mills theory. When |k+m=n|, the amplituhedron is isomorphic to the totally nonnegative Grassmannian, and when |k=1|, the amplituhedron is a cyclic polytope. While the case |m=4| is most relevant to physics, the amplituhedron is an interesting mathematical object for any |m|. In this article, we study it in the case |m=1|. We start by taking an orthogonal point of view and define a related “B-amplituhedron” |Bn,k,m|, which we show is isomorphic to |An,k,m|. We use this reformulation to describe the amplituhedron in terms of sign variation. We then give a cell decomposition of the amplituhedron |An,k,1| using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also show that |An,k,1| can be identified with the complex of bounded faces of a cyclic hyperplane arrangement, and describe how its cells fit together. We deduce that |An,k,1| is homeomorphic to a ball.