Decompositions of amplituhedra

  • Steven N. Karp

    University of Michigan, Ann Arbor, USA
  • Lauren K. Williams

    University of California, Berkeley, USA
  • Yan X Zhang

    San José State University, USA
Decompositions of amplituhedra cover
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Abstract

The (tree) amplituhedron is the image in the Grassmannian of the totally nonnegative Grassmannian , 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 planar supersymmetric Yang–Mills theory. In the case relevant to physics (), there is a collection of recursively-defined -dimensional BCFW cells in , whose images conjecturally "triangulate" the amplituhedron – that is, their images are disjoint and cover a dense subset of . 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 , the images of these cells are disjoint in . We also conjecture that for arbitrary even , there is a decomposition of the amplituhedron involving precisely top-dimensional cells (of dimension ), where is the number of plane partitions contained in an box. This agrees with the fact that when , the number of BCFW cells is the Narayana number .

Cite this article

Steven N. Karp, Lauren K. Williams, Yan X Zhang, Decompositions of amplituhedra. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 7 (2020), no. 3, pp. 303–363

DOI 10.4171/AIHPD/87