The m=1 amplituhedron and cyclic hyperplane arrangements

Understand

The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,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.