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SUMMARY:The hypersimplex and the m=2 amplituhedron - Melissa Sherman-Benne
 tt (University of Michigan)
DTSTART:20211111T130000Z
DTEND:20211111T133000Z
UID:TALK164794@talks.cam.ac.uk
DESCRIPTION:I'll discuss a curious correspondence between the m=2 amplituh
 edron\, a 2k-dimensional subset of Gr(k\, k+2)\, and the hypersimplex\, an
  (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are
  both images of the totally nonnegative Grassmannian (under the amplituhed
 ron map and the moment map\, respectively)\, but are different dimensions 
 and live in very different ambient spaces. In joint work with Matteo Paris
 i and Lauren Williams\, we give a bijection between decompositions of the 
 amplituhedron into "postroid tiles" and decompositions of the hypersimplex
  into positroid polytopes (originally conjectured by Lukowski--Parisi--Wil
 liams). Along the way\, we find a cluster connection: the positroid tiles 
 are the positive parts of new cluster varieties in Gr(k\, k+2). We also pr
 ove a sign-flip description of the amplituhedron conjectured by Arkani-Ham
 ed--Thomas--Trnka and introduce a new\, finer decomposition of the amplitu
 hedron into Eulerian-number-many chambers.
LOCATION:Seminar Room 1\, Newton Institute
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