The Hypersimplex VS the Amplituhedron - Signs, Triangulations, Clusters and Eulerian Numbers

1 June 2021
Matteo Parisi

In this talk I will discuss a striking duality, T-duality, we discovered between two seemingly unrelated objects: the hypersimplex and the m=2 amplituhedron. We draw novel connections between them and prove many new properties. We exploit T-duality to relate their triangulations and generalised triangles (maximal cells in a triangulation). We subdivide the amplituhedron into chambers as the hypersimplex can be subdivided into simplices - both enumerated by Eulerian numbers. Along the way, we prove several conjectures on the amplituhedron and find novel cluster-algebraic structures, e.g. a generalisation of cluster adjacency.

This is based on the joint work with Lauren Williams and Melissa Sherman-Bennett

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