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{
    "pk": 64924,
    "title": "Harmonic differential forms for pseudo-reflection groups II. Bi-degree bounds",
    "subtitle": null,
    "abstract": "This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic \\(0\\). Our most general result determines the top component in total degree, which we prove for all Shephard-Todd groups \\(G(m, p, n)\\) with \\(m \\neq p\\) or \\(m=1\\). Our strongest result gives tight bi-degree bounds and is proven for all \\(G(m, 1, n)\\), which includes the Weyl groups of types \\(A\\) and \\(B\\)/\\(C\\). For symmetric groups (i.e. type \\(A\\)), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund-Remmel-Wilson. Finally, we examine analogues of a classic theorem of Steinberg and the Operator Theorem of Haiman.\nOur arguments build on the type-independent classification of semi-invariant harmonic differential forms carried out in the first paper in this sequence. In this paper we use concrete constructions including Gröbner and Artin bases for the classical coinvariant algebras of the pseudo-reflection groups \\(G(m, p, n)\\), which we describe in detail. We also prove that exterior differentiation is exact on the super-coinvariant algebra of a general pseudo-reflection group. Finally, we discuss related conjectures and enumerative consequences.\n \n \n \nMathematics Subject Classifications: 05E16 (Primary), 20F55, 05A15 (Secondary)\n \nKeywords: Coinvariant algebras, pseudo-reflection groups, Gröbner basis, Artin basis, differential forms, exterior derivatives",
    "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": "Coinvariant algebras"
        },
        {
            "word": "pseudo-reflection groups"
        },
        {
            "word": "Gröbner basis"
        },
        {
            "word": "Artin basis"
        },
        {
            "word": "differential forms"
        },
        {
            "word": "exterior derivatives"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/2289r0v1",
    "frozenauthors": [
        {
            "first_name": "Joshua",
            "middle_name": "P.",
            "last_name": "Swanson",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of Southern California, Los Angeles, CA, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Nolan",
            "middle_name": "R.",
            "last_name": "Wallach",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of California San Diego, La Jolla, CA, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2023-12-22T14:46:12Z",
    "date_accepted": "2023-12-22T14:46:12Z",
    "date_published": "2023-12-22T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64924/galley/49734/download/"
        }
    ]
}