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{
    "pk": 64985,
    "title": "Viennot shadows and graded module structure in colored permutation groups",
    "subtitle": null,
    "abstract": "Let \\(\\mathbf{x}_{n \\times n}\\) be a matrix of \\(n \\times n\\) variables, and let \\(\\mathbb{C}[\\mathbf{x}_{n \\times n}]\\) be the polynomial ring on these variables. Let \\(\\mathfrak{S}_{n,r}\\) be the group of colored permutations, consisting of \\({n \\times n}\\) complex matrices with exactly one nonzero entry in each row and column, where each nonzero entry is an \\(r\\)-th root of unity. We associate an ideal \\(I_{\\mathfrak{S}_{n,r}} \\subseteq \\mathbb{C}[\\mathbf{x}_{n \\times n}]\\) with the group \\(\\mathfrak{S}_{n,r}\\), and use orbit harmonics to give an ideal-theoretic extension of the Viennot shadow line construction to \\(\\mathfrak{S}_{n,r}\\). This extension gives a standard monomial basis of \\(\\mathbb{C}[\\mathbf{x}_{n \\times n}]/I_{\\mathfrak{S}_{n,r}}\\), and introduces an analogous definition of \"longest increasing subsequence\" to the group \\(\\mathfrak{S}_{n,r}\\). We examine the extension of Chen's conjecture to this analogy. We also study the structure of \\(\\mathbb{C}[\\mathbf{x}_{n \\times n}]/I_{\\mathfrak{S}_{n,r}}\\) as a graded \\(\\mathfrak{S}_{n,r} \\times \\mathfrak{S}_{n,r}\\) module, which subsequently induces a graded \\(\\mathfrak{S}_{n,r} \\times \\mathfrak{S}_{n,r}\\) module structure on the \\(\\mathbb{C}\\)-algebra \\(\\mathbb{C}[\\mathfrak{S}_{n,r}]\\).\n \nMathematics Subject Classifications: 05E10, 05E16, 05E18, 05E14\n \nKeywords: Viennot's shadow lines, orbit harmonics, ideals, graded modules",
    "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": "Viennot's shadow lines"
        },
        {
            "word": "orbit harmonics"
        },
        {
            "word": "ideals"
        },
        {
            "word": "graded modules"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/3xr6d7m1",
    "frozenauthors": [
        {
            "first_name": "Jasper",
            "middle_name": "Moxuan",
            "last_name": "Liu",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of California San Diego, California, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2025-07-15T20:11:05+05:00",
    "date_accepted": "2025-07-15T20:11:05+05:00",
    "date_published": "2025-07-15T12:00:00+05:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64985/galley/49795/download/"
        }
    ]
}