{
  "schema_version": 1,
  "problem_number": "OWR-14299292-011",
  "title": "Fixed-Point-Free Representations of Alternating Groups of Twin-Prime Degree: A Partial Answer to a Question of M. Lee",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "M. Lee asked at the Oberwolfach workshop \"Computational Group Theory\" (June 2025) whether, for a prime p such that p − 2 is also prime, the alternating group A_p has a faithful irreducible representation over a field of some characteristic r > 0 in which all elements of orders p and p − 2 act without nonzero fixed vectors, and remarked that this is easily seen to fail for r > p. We give a partial answer: yes for p = 5, no for p = 7, 13, 19 (13 and 19 conditional on published decomposition matrices); open for p >= 31. For p = 5 such a representation exists exactly in characteristic 2 (A_5 ≅ SL_2(4) on F_4^2). For p = 7 the answer is no in every characteristic, without tables, with explicit certificates in F_2A_7 and F_3A_7. For p = 13 with r = 2, 3, 5 and p = 19 with r = 2, 3, 5, 7 the answer rests on the decomposition matrices of symmetric groups published by the Modular Atlas project, generated from the character table library of GAP; the other characteristics are excluded without tables. For every such p the question reduces to S_p and to primes r < p/2: r = p and r = p − 2 are impossible, r > p is Lee's remark (its short proof also covers characteristic 0), and p/2 < r < p − 2 is excluded by a standard Brauer-tree argument, which for a p-cycle alone Zalesski attributes to Thompson. For odd r the question reduces further to the simple modules of S_{p−2} with a fixed-point-free (p − 2)-cycle, a case of an open problem of Zalesski. No printed answer to the question was found in the sources searched; a search that finds nothing is not a proof of novelty. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "keywords": [
    "alternating groups",
    "symmetric groups",
    "modular representations",
    "fixed-point-free action",
    "eigenvalue 1",
    "Brauer characters",
    "decomposition matrices",
    "Brauer trees",
    "twin primes",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-14299292-011",
    "math.RT",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-21",
  "publication_date": "2026-10-21",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-21",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299292-011/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299292-011/paper.pdf?v=3c18c7460371",
  "doi": "10.5281/zenodo.23528869",
  "zenodo_record_url": "https://zenodo.org/records/23528869",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The answers for p = 13 and 19 are conditional on published decomposition matrices (Modular Atlas); p = 7 has table-free certificates.",
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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."
}