{
  "abstract": "We derive the triangular prism equations (TPE) for near-group fusion rules of type $G+|G|$, where $G$ is a finite abelian group. We prove that spherical categorification with trivial Frobenius--Schur indicator over any field in which $|G|$ is invertible is equivalent to the solvability of the TPE system. For every cyclic group, we construct an explicit solution to the TPE system using hyperbolic gamma functions, yielding both a spherical category with negative simple-object dimensions and, after Galois conjugation, a pseudo-unitary category with positive dimensions. For scalar solutions with positive dimensions, we establish an exact singular-value formula and a uniform gap, which provide explicit sufficient criteria for unitarity. Finally, we show that the unitarity of the cyclic Galois-conjugated coefficients for arbitrary orders corresponds to a specialization of a real-multiplication conjecture implied by the order-one abelian Stark conjecture, revealing a deep connection between categorical unitarity and Hilbert's twelfth problem for real quadratic fields.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11652",
  "author_total": 5,
  "authors": [
    {
      "name": "Huixuan He"
    },
    {
      "name": "Zhengwei Liu"
    },
    {
      "name": "Fan Lu"
    },
    {
      "name": "Sebastien Palcoux"
    },
    {
      "name": "Yunxiang Ren"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07t6fbcx2109f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11652",
  "fulltext_url": null,
  "id": "07t6fbcx2109f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11652",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7221181874",
  "pdf_url": "https://arxiv.org/pdf/2610.11652",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07t6fbcx2109f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11652",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11652"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Triangular prism equations for near-group categories: algebraic solutions and a unitary conjecture",
  "title_text": "Triangular prism equations for near-group categories: algebraic solutions and a unitary conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/07t6fbcx2109f",
  "venue": "arXiv",
  "versions": []
}