{
  "abstract": "We study the Ooms multiplicities of Frobenius maximal parabolics $W(a,b)$ of type~$A$. For $a=mb+r$ with $1\\le r<b$, the Euclidean algorithm relates their potential histograms to those of the smaller algebra $W(r,b-r)$. We prove strict unimodality for every positive coprime pair: the multiplicities increase strictly to the equal central values at eigenvalues $0$ and $1$, then decrease strictly. This strengthens the recent maximal-parabolic unimodality theorem of Giaquinto, Irving, Lauve, and Mastnak. We also prove log-concavity for the unbounded families $W(mb\\pm t,b)$, where $t\\in\\{1,2,3\\}$, $b>t$, $\\gcd(b,t)=1$, and $m\\ge1$. These results cover every Frobenius case with smaller block at most $8$, and also smaller block $10$. For each fixed residue class, we then prove that log-concavity of the whole family is determined by finitely many initial spectra. The finite cutoff is obtained symbolically. Exact integer verification of the resulting finite lists proves strict internal log-concavity whenever the smaller block is at most $128$, with no bound on the larger block. The unrestricted log-concavity conjecture is not proved here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27153",
  "author_total": 1,
  "authors": [
    {
      "name": "Vincent E. Coll Jr"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9tmghtvtm5s0e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27153",
  "fulltext_url": null,
  "id": "9tmghtvtm5s0e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27153",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214241452",
  "pdf_url": "https://arxiv.org/pdf/2609.27153",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9tmghtvtm5s0e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27153",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27153"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity",
  "title_text": "Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/9tmghtvtm5s0e",
  "venue": "arXiv",
  "versions": []
}