{"pk":64844,"title":"The hull metric on Coxeter groups","subtitle":null,"abstract":"We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for $B_n$ (and new proof for $\\mathfrak{S}_n$) use a combinatorial insertion map closely related to the well-studied promotion operator on linear extensions; this map may be of independent interest.  We also note that the inequalities in question can be interpreted as a triangle inequalities, so that convex hulls can be used to define a new invariant metric on $W$ whenever our conjecture holds. Geometric properties of this metric are an interesting direction for future research.\n \nMathematics Subject Classifications: 05A20, 05C12, 05E16, 20F55\n \nKeywords: Linear extension, promotion, Coxeter group, convex hull, metric","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":"Linear extension"},{"word":"promotion"},{"word":"Coxeter group"},{"word":"convex hull"},{"word":"metric"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/0hz0g20z","frozenauthors":[{"first_name":"Christian","middle_name":"","last_name":"Gaetz","name_suffix":"","institution":"Department of Mathematics, Cornell University, Ithaca, NY, U.S.A.","department":""},{"first_name":"Yibo","middle_name":"","last_name":"Gao","name_suffix":"","institution":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.","department":""}],"date_submitted":"2022-06-25T16:00:26Z","date_accepted":"2022-06-25T16:00:26Z","date_published":"2022-06-30T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64844/galley/49654/download/"}]}