Decompositions of amplituhedra

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. In the case relevant to physics (m=4), there is a collection of recursively-defined 4k-dimensional BCFW cells in the totally nonnegative part of Gr(k,n), whose images conjecturally "triangulate" the amplituhedron--that is, their images are disjoint and cover a dense subset of A(n,k,4). In this paper, we approach this problem by first giving an explicit (as opposed to recursive) description of the BCFW cells. We then develop sign-variational tools which we use to prove that when k=2, the images of these cells are disjoint in A(n,k,4). We also conjecture that for arbitrary even m, there is a decomposition of the amplituhedron A(n,k,m) involving precisely M(k, n-k-m, m/2) top-dimensional cells (of dimension km), where M(a,b,c) is the number of plane partitions contained in an a x b x c box. This agrees with the fact that when m=4, the number of BCFW cells is the Narayana number N(n-3, k+1).