Séminaire Lotharingien de Combinatoire, 91B.85 (2024), 12 pp.

Chaim Even-Zohar, Tsviqa Lakrec, Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler and Lauren Williams

Cluster Algebras and Tilings for The m = 4 Amplituhedron

Abstract. The amplituhedron AZn,k,m is the image of the positive Grassmannian Grk,n≥ 0 under the map Z~: Grk,n≥ 0 -> Grk,k+m induced by a positive linear map Z : Rn -> Rk+m. It was originally introduced in physics in order to give a geometric interpretation of scattering amplitudes. More specifically, one can compute scattering amplitudes in N=4 SYM by `tiling' the m=4 amplituhedron AZn,k,4 - that is, decomposing AZn,k,4 into `tiles' (closures of images of 4k-dimensional cells of Grk,n≥ 0 on which Z~ is injective). In this article we deepen both our understanding of tiles and tilings of the m=4 amplituhedron and the connection with cluster algebras. Firstly, we prove the cluster adjacency conjecture for BCFW tiles of AZn,k,4, which says that facets of tiles are cut out by collections of compatible cluster variables for Gr4,n. Secondly, we describe each BCFW tile as the semialgebraic set in Grk,k+4 where certain cluster variables have particular signs. Finally, we prove the BCFW tiling conjecture, which says that any way of iterating the BCFW recurrence gives rise to a tiling of the amplituhedron AZn,k,4. Along the way, we introduce a method to construct seeds for Gr4,n comprised of high-degree cluster variables, which may be of independent interest in the study of cluster algebras.


Received: November 15, 2023. Accepted: February 15, 2024. Final version: April 1, 2024.

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