Longer nilpotent series for classical unipotent subgroups

TL;DR

This paper extends the analysis of nilpotent groups by introducing longer, more refined series for classical unipotent subgroups, revealing detailed structure and growth patterns in their associated graded Lie rings.

Contribution

It computes the adjoint refinements of the lower central series for classical Chevalley group unipotent subgroups over finite fields, establishing their length and properties.

Findings

Series length is Θ(d^2) for classical types.

Nearly all factors have p-bounded order.

Provides detailed structure of associated graded Lie rings.

Abstract

In studying nilpotent groups, the lower central series and other variations can be used to construct an associated $\mathbb{Z}^+$-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized substantially by introducing $\mathbb{N}^d$-graded Lie rings. We compute the adjoint refinements of the lower central series of the unipotent subgroups of the classical Chevalley groups over the field $\mathbb{Z}/p\mathbb{Z}$ of rank $d$. We prove that, for all the classical types, this characteristic filter is a series of length $\Theta(d^2)$ with nearly all factors having $p$-bounded order.