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{
    "pk": 64949,
    "title": "Combinatorial descriptions of biclosed sets in affine type",
    "subtitle": null,
    "abstract": "Let \\(W\\) be a Coxeter group and let \\(\\Phi^+\\) be the positive roots. A subset \\(B\\) of \\(\\Phi^+\\) is called \"biclosed\" if, whenever we have roots \\(\\alpha\\), \\(\\beta\\) and \\(\\gamma\\) with \\(\\gamma \\in \\mathbb{R}_{›0} \\alpha + \\mathbb{R}_{›0} \\beta\\), if \\(\\alpha\\) and \\(\\beta \\in B\\) then \\(\\gamma \\in B\\) and, if \\(\\alpha\\) and \\(\\beta \\not\\in B\\), then \\(\\gamma \\not\\in B\\). The finite biclosed sets are the inversion sets of the elements of \\(W\\), and the containment between finite inversion sets is the weak order on \\(W\\). Dyer suggested studying the poset of all biclosed subsets of \\(\\Phi^+\\), ordered by containment, and conjectured that it is a complete lattice. As progress towards Dyer's conjecture, we classify all biclosed sets in the affine root systems. We provide both a type uniform description, and concrete models in the classical types \\(\\widetilde{A}\\), \\(\\widetilde{B}\\), \\(\\widetilde{C}\\), \\(\\widetilde{D}\\). We use our models to prove that biclosed sets form a complete lattice in types \\(\\widetilde{A}\\) and \\(\\widetilde{C}\\), and we classify which biclosed sets are separable and which are weakly separable.\n \nMathematics Subject Classifications: 20F55, 17B22, 06B23\n \nKeywords: Coxeter groups, root systems, affine Coxeter groups, lattice theory",
    "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": "Coxeter groups"
        },
        {
            "word": "root systems"
        },
        {
            "word": "affine Coxeter groups"
        },
        {
            "word": "lattice theory"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/9v68j10d",
    "frozenauthors": [
        {
            "first_name": "Grant",
            "middle_name": "T.",
            "last_name": "Barkley",
            "name_suffix": "",
            "institution": "Department of Mathematics, Harvard University, Cambridge, MA, U.S.A.",
            "department": ""
        },
        {
            "first_name": "David",
            "middle_name": "E",
            "last_name": "Speyer",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of Michigan, Ann Arbor, MI, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2024-09-26T15:53:29+02:00",
    "date_accepted": "2024-09-26T15:53:29+02:00",
    "date_published": "2024-09-30T09:00:00+02:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64949/galley/49759/download/"
        }
    ]
}