{"pk":64877,"title":"Shelling the \\(m=1\\) amplituhedron","subtitle":null,"abstract":"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.\n \nMathematics Subject Classifications: 06A07, 14M15, 81T60, 05A19\n \nKeywords: Amplituhedron, shellability, Eulerian number, log concavity, Sperner property","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Amplituhedron"},{"word":"shellability"},{"word":"Eulerian number"},{"word":"log concavity"},{"word":"Sperner property"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/245432bz","frozenauthors":[{"first_name":"Steven","middle_name":"N.","last_name":"Karp","name_suffix":"","institution":"Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, U.S.A.","department":""},{"first_name":"John","middle_name":"","last_name":"Machacek","name_suffix":"","institution":"Department of Mathematics, University of Oregon, Eugene, Oregon, U.S.A.","department":""}],"date_submitted":"2023-03-14T15:27:58Z","date_accepted":"2023-03-14T15:27:58Z","date_published":"2023-03-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64877/galley/49687/download/"}]}