Séminaire Lotharingien de Combinatoire, 86B.31 (2022), 12 pp.
Matteo Parisi, Melissa Sherman-Bennett and Lauren Williams
The m=2 Amplituhedron and the Hypersimplex
Abstract.
The hypersimplex Δk+1,n is the image of the positive
Grassmannian Gr≥0k+1,n under the moment map.
It is a polytope of dimension n-1 in Rn.
Meanwhile, the amplituhedron
A is the image of Gr≥0k,n under an amplituhedron map Z~ induced by
a positive matrix Z.
Introduced in the context of scattering amplitudes, it is not a polytope, and
is a full dimensional subset of Grk,k+2.
Nevertheless, there seem to be remarkable connections between these two objects,
as conjectured by Lukowski-Parisi-Williams (LPW).
We use ideas from oriented matroid theory, total positivity,
and the geometry of the hypersimplex and positroid polytopes
to obtain a deeper understanding of the amplituhedron.
We show that the inequalities cutting out positroid polytopes - moment map images of positroid cells - translate into sign conditions cutting out Grasstopes - amplituhedron map images of positroid cells.
Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices - with both chambers and
simplices enumerated by the Eulerian numbers.
We use these properties to prove the main conjecture of (LPW):
a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron A for allZ.
We also prove
Arkani-Hamed-Thomas-Trnka's conjectural sign-flip characterization of A.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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