A new look at the decomposition of unipotents and the normal structure of Chevalley groups

TL;DR

This paper investigates the decomposition of unipotent elements in Chevalley groups over rings, providing new methods to prove normality of elementary subgroups by constructing specific 'good' elements, advancing understanding of their structure.

Contribution

It introduces a novel approach to decompose unipotents in Chevalley groups using 'good' elements, simplifying proofs of normality and the group's structure.

Findings

Decomposition of unipotents implies normality of elementary subgroups.

Constructing a single 'good' element suffices for normal structure proofs.

Further work will explore whether 'good' elements generate the entire elementary group.

Abstract

The current article continues a series of papers on decomposition of unipotents and its applications. Let $G(\Phi,R)$ be a Chevalley group with a reduced irreducible root system $\Phi$ over a commutative ring $R$. Fix $h\in G(\Phi,R)$. Call an element $a\in G(\Phi,R)$ "good", if it lies in the unipotent radical of a parabolic subgroup whereas the conjugate to $a$ by $h$ belongs to another proper parabolic subgroup (here we assume that all parabolics contain a given split maximal torus). Decomposition of unipotents is a representation of a root unipotent element as a product of "good" elements. Existence of such a decomposition implies a simple proof of the normality of the elementary subgroup and description of the normal structure of $G(\Phi,R)$. However, the decomposition is available not for all root systems. In the current article we show that for the proof of the standard normal structure it suffices to construct only one "good" element for the generic element of the scheme $G(\Phi,_-)$, and construct such a "good" element. The question whether "good" elements generate the whole elementary group will be addressed in a separate paper (the first version of the article was in Russian).