Séminaire Lotharingien de Combinatoire, 78B.20 (2017), 12 pp.
Steven N. Karp and Lauren K. Williams
The m=1 Amplituhedron and Cyclic Hyperplane Arrangements
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. 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
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. We deduce
that An,k,1 is homeomorphic to a ball.
Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.
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