Extremal Derived Length in the Unipotent Radicals of Cartan-Type Automorphism Groups
Abstract
Over perfect fields of characteristic p≥5, the unipotent radicals of the height-one special, Witt-Jacobson, and Hamiltonian automorphism groups in at least three, two, and four variables, respectively, attain the largest derived length allowed by their nilpotency classes. More strongly, every term of each derived series equals the corresponding power-of-two term of the lower central series. These equalities hold on finite-field points, with explicit Frattini quotients and lower p-central series. For the special family, a Cartier quotient measures the difference between the congruence and lower central filtrations. The Hamiltonian correction comes from flux and the socle line; the Witt-Jacobson family has none. We also identify the contact automorphism group scheme and, at arbitrary divided-power height, compute the Cartier-Verschiebung maps on restricted derivations and the Verschiebung filtration and distribution algebra of the Frobenius thickening of the special radical.
Related
- Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical Groups
- Algebraic derivations and radicals of automorphism groups of algebras
- An automorphism tower of length κ+
- Determination of weight enumerators of binary extremal self-dual codes of length 60
- Extremal graphs without cycles of length 0 modulo 4
BibTeX
@misc{ma2026extremal,
title = {{Extremal Derived Length in the Unipotent Radicals of Cartan-Type Automorphism Groups}},
author = {Chao Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.31711},
eprint = {2609.31711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31711},
}
RIS
TY - UNPB
TI - Extremal Derived Length in the Unipotent Radicals of Cartan-Type Automorphism Groups
AU - Chao Ma
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.31711
UR - https://arxiv.org/abs/2609.31711
AB - Over perfect fields of characteristic p≥5, the unipotent radicals of the height-one special, Witt-Jacobson, and Hamiltonian automorphism groups in at least three, two, and four variables, respectively, attain the largest derived length allowed by their nilpotency classes. More strongly, every term of each derived series equals the corresponding power-of-two term of the lower central series. These equalities hold on finite-field points, with explicit Frattini quotients and lower p-central series. For the special family, a Cartier quotient measures the difference between the congruence and lower central filtrations. The Hamiltonian correction comes from flux and the socle line; the Witt-Jacobson family has none. We also identify the contact automorphism group scheme and, at arbitrary divided-power height, compute the Cartier-Verschiebung maps on restricted derivations and the Verschiebung filtration and distribution algebra of the Frobenius thickening of the special radical.
ER -