{
  "abstract": "The exceptional group $F_4(\\mathbb{C})$ has two conjugacy classes of involutions: an involution $t$ with centralizer $\\mathrm{Spin}(9,\\mathbb{C})$, and the principal involution, with centralizer $(\\mathrm{SL}(2,\\mathbb{C})\\times\\mathrm{Sp}(6,\\mathbb{C}))/\\{\\pm1\\}$. We determine the values of all irreducible characters of $F_4(\\mathbb{C})$ at $t$. For the irreducible representation with highest weight $aω_1+bω_2+cω_3+dω_4$ (in Bourbaki's labelling), we show that the character vanishes at $t$ if and only if $c$ and $d$ are both odd. In all other cases its value is $\\pm 2^{-7}$ times the dimension of an explicit irreducible representation of $\\mathrm{Spin}(9,\\mathbb{C})$. The proof is a limiting argument in the Weyl character formula, combined with a decomposition over the three cosets of $W(B_4)$ in $W(F_4)$ and the triality action of $W(F_4)=W(D_4)\\rtimes S_3$ on the weight lattice modulo the $D_4$-lattice. At the principal involution, Nadimpalli, Pattanayak and Prasad showed that the character value is $\\pm 1/2$ times the dimension of an irreducible representation of $\\mathrm{SL}(2,\\mathbb{C})\\times\\mathrm{Spin}(7,\\mathbb{C})$. Comparing the two results, we find that the characters of $F_4(\\mathbb{C})$ vanish at the two involutions for exactly the same highest weights. Together these results determine all irreducible characters of $F_4(\\mathbb{C})$ at all involutions. We also explain how our formula at $t$ relates to the generalized Weyl character formula of Gross, Kostant, Ramond and Sternberg.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05116",
  "author_total": 1,
  "authors": [
    {
      "name": "CHAYAN KARMAKAR"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0yh893mvy80yd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05116",
  "fulltext_url": null,
  "id": "0yh893mvy80yd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05116",
  "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": "W7220737401",
  "pdf_url": "https://arxiv.org/pdf/2610.05116",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0yh893mvy80yd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05116",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05116"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Characters of $F_4(\\mathbb{C})$ at involutions",
  "title_text": "Characters of F_4(ℂ) at involutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/0yh893mvy80yd",
  "venue": "arXiv",
  "versions": []
}