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Shelling the \(m=1\) amplituhedron

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The amplituhedron \(\mathcal{A}_{n,k,m}\) was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar \(\mathcal{N}=4\) supersymmetric Yang-Mills theory. It is a projection inside the Grassmannian \(\text{Gr}_{k,k+m}\) of the totally nonnegative part of \(\text{Gr}_{k,n}\). Karp and Williams (2019) studied the \(m=1\) amplituhedron \(\mathcal{A}_{n,k,1}\), giving a regular CW decomposition of it. Its face poset \(R_{n,l}\) (with \(l := n-k-1\)) consists of all projective sign vectors of length \(n\) with exactly \(l\) sign changes. We show that \(R_{n,l}\) is EL-shellable, resolving a problem posed by Karp and Williams. This gives a new proof that \(\mathcal{A}_{n,k,1}\) is homeomorphic to a closed ball, which was originally proved by Karp and Williams. We also give explicit formulas for the \(f\)-vector and \(h\)-vector of \(R_{n,l}\), and show that it is rank-log-concave and strongly Sperner. Finally, we consider a related poset \(P_{n,l}\) introduced by Machacek (2019), consisting of all projective sign vectors of length \(n\) with at most \(l\) sign changes. We show that it is rank-log-concave, and conjecture that it is Sperner.

Mathematics Subject Classifications: 06A07, 14M15, 81T60, 05A19

Keywords: Amplituhedron, shellability, Eulerian number, log concavity, Sperner property

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