Nonunimodal principal spectra of Frobenius parabolic Lie algebras
Abstract
Strict unimodality of Ooms spectra fails for Frobenius parabolic Lie algebras beyond the maximal-parabolic class. We construct explicit Frobenius parabolics whose principal multiplicity sequences contain arbitrarily long outward rises: for every L≥1 there are L consecutive negative first differences, each of quadratic size in L. In a simple one-parameter subfamily the transition to nonunimodality occurs in slie136, where two adjacent multiplicities are 541<542. We also give a representation-theoretic criterion for unimodality. If F is a Frobenius functional with semisimple integral principal element H and g=⨁jgj, the Kirillov form makes the centered grading symplectic and its character has the unique expansion Θg(x)=∑m≥1(dimgm−dimgm+1) χ2m−1(x) in irreducible SL2 characters. Thus Ooms unimodality is equivalent to effectivity of this virtual SL2 character, or equivalently to extension of the centered principal cocharacter to a symplectic SL2 action. For Frobenius seaweeds the unbroken-spectrum theorem says that, after centering, every odd weight in the support occurs; strict unimodality asks for the stronger positivity of every irreducible coefficient. A companion paper proves that stronger property for every Frobenius maximal parabolic. The families constructed here show that beyond that class there can be arbitrarily long blocks of negative irreducible coefficients, with unbounded total negative mass.
Related
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BibTeX
@misc{jr2026nonunimodal,
title = {{Nonunimodal principal spectra of Frobenius parabolic Lie algebras}},
author = {Vincent E. Coll Jr},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33173},
eprint = {2609.33173},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33173},
note = {CC0},
}
RIS
TY - UNPB
TI - Nonunimodal principal spectra of Frobenius parabolic Lie algebras
AU - Vincent E. Coll Jr
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33173
UR - https://arxiv.org/abs/2609.33173
AB - Strict unimodality of Ooms spectra fails for Frobenius parabolic Lie algebras beyond the maximal-parabolic class. We construct explicit Frobenius parabolics whose principal multiplicity sequences contain arbitrarily long outward rises: for every L≥1 there are L consecutive negative first differences, each of quadratic size in L. In a simple one-parameter subfamily the transition to nonunimodality occurs in slie_136, where two adjacent multiplicities are 541<542. We also give a representation-theoretic criterion for unimodality. If F is a Frobenius functional with semisimple integral principal element H and g=⨁_jg_j, the Kirillov form makes the centered grading symplectic and its character has the unique expansion Θ_g(x)=∑_m≥1(dimg_m−dimg_m+1) χ_2m−1(x) in irreducible SL_2 characters. Thus Ooms unimodality is equivalent to effectivity of this virtual SL_2 character, or equivalently to extension of the centered principal cocharacter to a symplectic SL_2 action. For Frobenius seaweeds the unbroken-spectrum theorem says that, after centering, every odd weight in the support occurs; strict unimodality asks for the stronger positivity of every irreducible coefficient. A companion paper proves that stronger property for every Frobenius maximal parabolic. The families constructed here show that beyond that class there can be arbitrarily long blocks of negative irreducible coefficients, with unbounded total negative mass.
ER -