{
  "abstract": "Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\\otimes k}$, counted with multiplicity, and let $δ_p=1-\\log_{p^2}\\!\\bigl(\\tfrac{p+1}{2}\\bigr)$. Then there exists a smooth, strictly positive, multiplicatively $p^2$-periodic function $ω(t)$ such that $b_k$ is asymptotic to $ω(k)k^{-δ_p}2^k$. We also show that $t^{-δ}ω(t)$ arises as the limiting density of renormalized convolutions of rescaled copies of a positive weight $3/2$ theta function, obtained from the boundary heat flux of the Dirichlet heat kernel on $(0,p)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37706",
  "author_total": 2,
  "authors": [
    {
      "name": "Aranya Lahiri"
    },
    {
      "name": "Nai-Heng Sheu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9jtmc9m2kkh0h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37706",
  "fulltext_url": null,
  "id": "9jtmc9m2kkh0h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37706",
  "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.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214968807",
  "pdf_url": "https://arxiv.org/pdf/2609.37706",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9jtmc9m2kkh0h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2609.37706",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37706"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Asymptotics for number of indecomposable components of tensor powers of the natural $\\mathrm{SL}_2$-module in odd characteristic",
  "title_text": "Asymptotics for number of indecomposable components of tensor powers of the natural SL_2-module in odd characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/9jtmc9m2kkh0h",
  "venue": "arXiv",
  "versions": []
}