{
  "abstract": "Let $G=UT_3(\\F_q)$ be the finite Heisenberg group. We study the image closures and secant varieties of a family of equivariant bilinear maps between representations induced from central characters. Evaluation at a basepoint identifies the ambient space with $M_{q^2}(\\C)$. A single channel is represented by $A^T\\operatorname{diag}(d)B$, with $A$ and $B$ in prescribed twisted-convolution algebras. We prove that its affine image closure has dimension $3q^2-2$ for every central-character pair and that its affine second-secant closure has the expected dimension for every prime power $q$. For $q\\ge3$, the proof constructs two differential images with zero intersection. We also prove a subgroup-lifting theorem for channel parameterizations on finite abelian groups with bicharacter coefficients: a parameter point with differential rank $r(3|H|-2)$ on a subgroup $H$ extends by zero to a parameter point with differential rank $r(3|K|-2)$ on the ambient group $K$, provided that every parameter vector is nonzero. Exact Jacobian certificates determine all secant dimensions for $q=2,3,4,5$, with generic widths $2,4,6,9$. The lifting theorem yields specified higher-secant dimensions on infinite families of extension fields. Attainment of the dimension lower bound for arbitrary $q$ remains a conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01104",
  "author_total": 2,
  "authors": [
    {
      "name": "Nguyen Bich Van"
    },
    {
      "name": "Nguyen Trung Kien"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8r1z33pa4q5va.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01104",
  "fulltext_url": null,
  "id": "8r1z33pa4q5va",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01104",
  "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.AG"
    ],
    [
      "arxiv",
      "math.GR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01104",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8r1z33pa4q5va.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01104",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01104"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Secant dimensions and bilinear decomposition widths for finite Heisenberg groups",
  "title_text": "Secant dimensions and bilinear decomposition widths for finite Heisenberg groups",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/8r1z33pa4q5va",
  "venue": "arXiv",
  "versions": []
}