{
  "abstract": "Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\\mathbb E\\colon P\\times N\\to \\mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\\widehat G$. We show then that such representations are irreducible and the corresponding ordinary and dual $2$-cocycles are nondegenerate if and only if the matrix $(\\mathbb E(p,n))_{p,n}$ is invertible, if and only if it is Hadamard. We illustrate the result with examples arising from symplectic nilpotent Lie algebras over finite fields with two complementary Lagrangian subalgebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11785",
  "author_total": 4,
  "authors": [
    {
      "name": "Pierre Bieliavsky"
    },
    {
      "name": "Victor Gayral"
    },
    {
      "name": "Sergey Neshveyev"
    },
    {
      "name": "Lars Tuset"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2m0dcn013x1vs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11785",
  "fulltext_url": null,
  "id": "2m0dcn013x1vs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11785",
  "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.CO"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7221102131",
  "pdf_url": "https://arxiv.org/pdf/2610.11785",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2m0dcn013x1vs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.11785",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11785"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Facets of nondegeneracy: central type factor groups, Heisenberg representations and Hadamard matrices",
  "title_text": "Facets of nondegeneracy: central type factor groups, Heisenberg representations and Hadamard matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/2m0dcn013x1vs",
  "venue": "arXiv",
  "versions": []
}