{
  "abstract": "Let $k$ be an algebraically closed field of characteristic zero and $G$ a finite group. We realize the based Grothendieck ring $R_G$ of ${\\mathcal{R}\\mathit{ep}}(D(G))$-module categories as the degree-two cocycle-decorated double Burnside ring and derive an explicit Clifford formula for multiplication and for the action of $R_G$ on the Grothendieck group of ${\\operatorname{\\operatorname{\\mathsf{Vec}}}}_G$-module categories. We determine the extremal based ideals, study factorization through smaller groups, and prove a Mackey theorem for standard two-sided subgroup inductions. We prove that $\\mathbb C\\otimes_{\\mathbb Z}R_G$ is semisimple exactly when $G$ is cyclic. We study the Brauer--Picard action on indecomposable ${\\operatorname{\\operatorname{\\mathsf{Vec}}}}_G$-module categories and show that it is transitive exactly when $G$ is abelian of square-free exponent. For abelian $G$, we determine the possible abstract group types of Lagrangian subgroups of $G\\oplus\\widehat G$, apply the known orthogonal classification in the homocyclic case, and exhibit same-type nonconjugate Lagrangians for mixed exponents.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27020",
  "author_total": 1,
  "authors": [
    {
      "name": "Dmitri Nikshych"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/96srxdxk7mgwr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27020",
  "fulltext_url": null,
  "id": "96srxdxk7mgwr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27020",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7214211873",
  "pdf_url": "https://arxiv.org/pdf/2609.27020",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/96srxdxk7mgwr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27020",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27020"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Grothendieck Rings of Module Categories over Drinfeld Doubles",
  "title_text": "Grothendieck Rings of Module Categories over Drinfeld Doubles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/96srxdxk7mgwr",
  "venue": "arXiv",
  "versions": []
}