The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers

Abstract

The hypersimplex Δ k + 1 , n \Delta _{k+1,n} is the image of the positive Grassmannian G r k + 1 , n ≥ 0 Gr^{\geq 0}_{k+1,n} under the moment map. It is a polytope of dimension n − 1 n-1 in R n \mathbb {R}^n . Meanwhile, the amplituhedron A n , k , 2 ( Z ) \mathcal {A}_{n,k,2}(Z) is the projection of the positive Grassmannian G r k , n ≥ 0 Gr^{\geq 0}_{k,n} into the Grassmannian G r k , k + 2 Gr_{k,k+2} under a map Z ~ \tilde {Z} induced by a positive matrix Z ∈ M a t n , k + 2 > 0 Z\in Mat_{n,k+2}^{>0} . Introduced in the context of scattering amplitudes, it is not a polytope, and has full dimension 2 k 2k inside G r k , k + 2 Gr_{k,k+2} . Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper 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—images of positroid cells of G r k + 1 , n ≥ 0 Gr^{\geq 0}_{k+1,n} under the moment map—translate into sign conditions characterizing the T-dual Grasstopes—images of positroid cells of G r k , n ≥ 0 Gr^{\geq 0}_{k,n} under Z ~ \tilde {Z} . 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