A cluster of results on amplituhedron tiles

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in 𝒩=4 super Yang Mills theory. It generalizes cyclic polytopes and the positive Grassmannian, and has a very rich combinatorics with connections to cluster algebras. In this article we provide a series of results about tiles and tilings of the m=4 amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for _4,n. Secondly, we exhibit a tiling of the m=4 amplituhedron which involves a tile which does not come from the BCFW recurrence – the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for _4,n. This paper is a companion to our previous paper “Cluster algebras and tilings for the m=4 amplituhedron”.