{"pk":64859,"title":"Schubert polynomials as projections of Minkowski sums of Gelfand-Tsetlin polytopes","subtitle":null,"abstract":"Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of \\(\\mathfrak{gl}_n(\\mathbb{C})\\). The integer point transform of the Gelfand-Tsetlin polytope \\(\\mathrm{GT}(\\lambda)\\) projects to the Schur function \\(s_{\\lambda}\\). Schur functions form a distinguished basis of the ring of symmetric functions; they are also special cases of Schubert polynomials \\(\\mathfrak{S}_{w}\\) corresponding to Grassmannian permutations. For any permutation \\(w \\in S_n\\) with column-convex Rothe diagram, we construct a polytope \\(\\mathcal{P}_{w}\\) whose integer point transform projects to the Schubert polynomial \\(\\mathfrak{S}_{w}\\). Such a construction has been sought after at least since the construction of twisted cubes by Grossberg and Karshon in 1994, whose integer point transforms project to Schubert polynomials \\(\\mathfrak{S}_{w}\\) for all \\(w \\in S_n\\). However, twisted cubes are not honest polytopes; rather one can think of them as signed polytopal complexes. Our polytope \\(\\mathcal{P}_{w}\\) is a convex polytope, namely it is a Minkowski sum of Gelfand-Tsetlin polytopes of varying sizes. When the permutation \\(w\\) is Grassmannian, the Gelfand-Tsetlin polytope is recovered. We conclude by showing that the Gelfand-Tsetlin polytope is a flow polytope.\n \nMathematics Subject Classifications: 05E05\n \nKeywords: Schubert polynomials, Gelfand-Tsetlin polytopes, flow polytopes","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":"Schubert polynomials"},{"word":"Gelfand-Tsetlin polytopes"},{"word":"flow polytopes"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/4c6659v6","frozenauthors":[{"first_name":"Ricky","middle_name":"Ini","last_name":"Liu","name_suffix":"","institution":"Department of Mathematics, University of Washington, Seattle, WA, U.S.A.","department":""},{"first_name":"Karola","middle_name":"","last_name":"Mészáros","name_suffix":"","institution":"Department of Mathematics, Cornell University, Ithaca NY, U.S.A.","department":""},{"first_name":"Avery","middle_name":"St.","last_name":"Dizier","name_suffix":"","institution":"Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, U.S.A.","department":""}],"date_submitted":"2022-10-11T15:50:32Z","date_accepted":"2022-10-11T15:50:32Z","date_published":"2022-10-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64859/galley/49669/download/"}]}