{
  "schema_version": 1,
  "problem_number": "AIM-REPRESENTATION_THEORY-0102",
  "title": "A Degree-24 Obstruction to Hopf Structures on Symmetric-Group Supercharacter Spaces",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Consider the supercharacter theory of S_n whose superclasses are the S_n-conjugacy classes contained in A_n, together with the single block S_n-A_n. We prove that its superclass-function spaces, summed over n with one-dimensional components in degrees zero and one, admit no connected graded Hopf algebra structure over a field of characteristic zero. The argument uses a known nonnegative Euler-product condition for Hopf Hilbert series. The first negative Euler exponent occurs in degree 24 and equals -1: lower degrees force 795 basis monomials, but the required dimension is 794. The two coarser natural supercharacter families fail the same test in degree 4. These obstructions exclude arbitrary graded Hopf operations, not only quotients of symmetric functions. We state the families explicitly and do not claim a classification of all symmetric-group supercharacter theories.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.RT",
    "math.QA",
    "math.CO"
  ],
  "keywords": [
    "supercharacter theory",
    "symmetric group",
    "Hopf algebra",
    "Hilbert series",
    "Euler transform",
    "AIM-REPRESENTATION_THEORY-0102",
    "math.RT",
    "math.QA",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-representation-theory-0102/",
  "pdf_url": "https://eulersolve.org/papers/aim-representation-theory-0102/paper.pdf?v=132a14ebad39",
  "doi": "10.5281/zenodo.23018019",
  "zenodo_record_url": "https://zenodo.org/records/23018019",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The theorem concerns three explicitly stated supercharacter families with ordinary degree-n grading, one-dimensional degree-zero and degree-one components, and characteristic zero. The source does not name all four families in its question, so this is not claimed as a classification or a resolution of the whole source question. Signed, positive-characteristic, ungraded and regraded Hopf structures are not covered. The known Hilbert-series criterion is credited; self-audited, unrefereed, with no formal verification or absolute priority certificate claimed.",
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    "source.zip": {
      "sha256": "449d0c3bc52ce78844dd83628423a2cd2f0cc43f7b8c525480d3f58515f27d7b"
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    "verification_report.md": {
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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."
}