Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity

Abstract

We study the Ooms multiplicities of Frobenius maximal parabolics W(a,b) of type~A. For a=mb+r with 1≤ 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± t,b), where t∈{1,2,3}, b>t, gcd(b,t)=1, and m≥1. 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.

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BibTeX RIS

BibTeX

@misc{jr2026ooms,
  title = {{Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity}},
  author = {Vincent E. Coll Jr},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27153},
  eprint = {2609.27153},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27153},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity
AU  - Vincent E. Coll Jr
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.27153
UR  - https://arxiv.org/abs/2609.27153
AB  - We study the Ooms multiplicities of Frobenius maximal parabolics W(a,b) of type~A. For a=mb+r with 1≤ 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± t,b), where t∈{1,2,3}, b>t, gcd(b,t)=1, and m≥1. 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.
ER  -

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