The hypersimplex and the m=2 amplituhedron: Eulerian numbers, sign flips, triangulations

Abstract: Physicists Arkhani-Hamed and Trnka introduced the amplituhedron to better understand scattering amplitudes in N=4 super Yang-Mills theory. The amplituhedron is the image of the totally nonnegative Grassmannian under the "amplituhedron map", which is induced by matrix multiplication. Examples of amplituhedra include cyclic polytopes, the totally nonnegative Grassmannian itself, and cyclic hyperplane arrangements. In general, the amplituhedron is not a polytope. However, Lukowski--Parisi--Williams noticed a mysterious connection between the m=2 amplituhedron and the hypersimplex, and conjectured a correspondence between their fine positroidal subdivisions. I'll discuss joint work with Matteo Parisi and Lauren Williams, in which we prove one direction of this correspondence. Along the way, we prove an intrinsic description of the m=2 amplituhedron conjectured by Arkhani-Hamed--Thomas--Trnka; give a decomposition of the m=2 amplituhedron into Eulerian number-many sign chambers, in direct analogy to a triangulation of the hypersimplex; and find new cluster varieties in the Grassmannian.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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