The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers
The hypersimplex Δ <#comment/> k + 1 , n \Delta _{k+1,n} is the image of the positive Grassmannian G r k + 1 , n ≥ <#comment/> 0 Gr^{\geq 0}_{k+1,n} under the moment map. It is a polytope of dimension n − <#comment/> 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 ≥ <#comment/> 0 Gr^{\geq 0}_{k,n} into the Grassmannian G r k , k + 2 Gr_{k,k+2} under a map Z ~ <#comment/> \tilde {Z} induced by a positive matrix Z ∈ <#comment/> 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 ≥ <#comment/> 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 ≥ <#comment/> 0 Gr^{\geq 0}_{k,n} under Z ~ <#comment/> \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 the main conjecture of Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]: a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron A n , k , 2 ( Z ) \mathcal {A}_{n,k,2}(Z) for all Z Z . Moreover, we prove Arkani-Hamed–Thomas–Trnka’s conjectural sign-flip characterization of A n , k , 2 \mathcal {A}_{n,k,2} , and Łukowski–Parisi–Spradlin–Volovich’s conjectures on m = 2 m=2 cluster adjacency and on positroid tiles for A n , k , 2 \mathcal {A}_{n,k,2} (images of 2 k 2k -dimensional positroid cells which map injectively into A n , k , 2 \mathcal {A}_{n,k,2} ). Finally, we introduce new cluster structures in the amplituhedron.