Decompositions of amplituhedra
Steven N. Karp, Lauren K. Williams, Yan X Zhang

TL;DR
This paper provides an explicit description of BCFW cells in the amplituhedron, proves their disjointness for k=2, and conjectures a general decomposition involving plane partitions for arbitrary even m.
Contribution
It introduces an explicit description of BCFW cells, proves their disjointness for k=2, and conjectures a new combinatorial decomposition of amplituhedra for even m.
Findings
Disjointness of BCFW cells for k=2 in A(n,k,4)
Explicit description of BCFW cells
Conjectured decomposition involving plane partitions for even m
Abstract
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).…
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