{
  "schema_version": 1,
  "problem_number": "OWR-8415341-012",
  "title": "A graphical group whose number of irreducible characters of degree p^3 is not a polynomial in p",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a finite simple graph Γ and an odd prime p, let G_Γ(F_p) be the largest group of class at most two and exponent dividing p generated by elements x_v, one for each vertex v, such that x_u and x_v commute whenever u and v are not adjacent. Rossmann asked (Oberwolfach Report 38/2021; Question 1.10 in Bull. London Math. Soc. 2022) how the number of irreducible characters of G_Γ(F_p) of degree p^i depends on p, and suspected that these functions might be wild. We show that they are not always polynomials. Let Γ_MK be the bipartite graph on the points and lines of the Möbius–Kantor configuration 8_3 in which a point is joined to the five lines not containing it. For every odd prime p, G_{Γ_MK}(F_p) has exactly p^10 (h(p) + (1 + (−3/p)) (p − 1)^15) irreducible characters of degree p^3, where h is an explicit integer polynomial of degree 15 obtained by an exact computation; the count is a polynomial on each residue class modulo 3, but on no set of primes containing infinitely many primes ≡ 1 and infinitely many ≡ 2 (mod 3), and this needs no computer. We use the description of character degrees by ranks due to O'Brien and Voll and the fact, classical over the complex numbers, that 8_3 can be realised exactly over the fields containing a root of t^2 − t + 1. The counts are polynomials for i = 1, and for i ≤ 2 if Γ is bipartite. Conditionally on the Mnëv–Sturmfels theorem as stated by Belkale and Brosnan, who sketch its proof, some count is not even a polynomial on residue classes (existence only). The general behaviour remains open. This is an unrefereed note.",
  "result_type": "DISPROOF_EXPLICIT",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "keywords": [
    "graphical groups",
    "p-groups of class two",
    "irreducible characters",
    "character degrees",
    "alternating matrices with support constraints",
    "q-rook numbers",
    "Möbius–Kantor configuration",
    "PORC",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-8415341-012",
    "math.GR",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-8415341-012/",
  "pdf_url": "https://eulersolve.org/papers/owr-8415341-012/paper.pdf?v=ef80ef4279f4",
  "doi": "10.5281/zenodo.23707514",
  "zenodo_record_url": "https://zenodo.org/records/23707514",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The implicit polynomiality question is answered no by an explicit graph; the general behaviour asked about stays open; the polynomial h is computed by exact arithmetic and checked independently; the non-PORC statement is conditional on a theorem whose proof is only sketched in print; unrefereed.",
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    "source.zip": {
      "sha256": "cb73a14f480646ae69a5b6624a37a54e77e9ef0eda21f444f38867c3bc98e4bb"
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    "verification_report.md": {
      "sha256": "4fe904560fa6d09b641d6803d875d4a2ed2bfc9676f8c787d8c61f1ea69a152e"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}