2021/04/16 by Matteo Parisi, Parisi, Matteo, Melissa Sherman-Bennett +4
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Amino Acid Enzymes and Metabolism #DNA and Biological Computing #Finite Group Theory Research #Noncommutative and Quantum Gravity Theories #hep-th #math-ph #math.AG #math.CO #math.MP
paper · pdf · doi:10.48550/arxiv.2104.08254
openalex publication_date 2021/04/16 · openalex created_date 2023/01/25 · openalex updated_date 2026/08/01
The hypersimplex \Δk+1,n is the image of the positive Grassmannian\nGr\≥ 0k+1,n under the moment map. It is a polytope of dimension n-1\nin \ℝn. Meanwhile, the amplituhedron \An,k,2(Z) is the\nprojection of the positive Grassmannian Gr\≥ 0k,n into Grk,k+2\nunder a map \Z induced by a matrix Z\∈ \Matn,k+2>0.\nIntroduced in the context of scattering amplitudes, it is not a polytope, and\nhas dimension 2k. Nevertheless, there seem to be remarkable connections\nbetween these two objects via T-duality, as was first noted by\nLukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented\nmatroid theory, total positivity, and the geometry of the hypersimplex and\npositroid polytopes to obtain a deeper understanding of the amplituhedron. We\nshow that the inequalities cutting out positroid polytopes -- images of\npositroid cells of Gr\≥ 0k+1,n under the moment map -- translate into\nsign conditions characterizing the T-dual Grasstopes -- images of positroid\ncells of Gr\≥ 0k,n under \Z. Moreover, we subdivide the\namplituhedron into chambers, just as the hypersimplex can be subdivided into\nsimplices, with both chambers and simplices enumerated by the Eulerian numbers.\nWe prove the main conjecture of (LPW): a collection of positroid polytopes is a\ntriangulation of \Δk+1, n if and only if the collection of T-dual\nGrasstopes is a triangulation of \An,k,2(Z) for all Z.\nMoreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip\ncharacterization of \An,k,2(Z), and\nLukowski--Parisi--Spradlin--Volovich's conjectures on m=2 cluster adjacency\nand on generalized triangles (images of 2k-dimensional positroid cells which\nmap injectively into \An,k,2(Z)). Finally, we introduce new\ncluster structures in the amplituhedron.\n