Unitriangular factorisations of Chevalley groups

TL;DR

This paper investigates minimal length factorizations of Chevalley groups into unipotent radicals, improving known bounds over rings of stable rank 1 and exploring implications under the Generalised Riemann Hypothesis.

Contribution

It extends the understanding of unitriangular factorizations of Chevalley groups, providing new bounds and connecting classical results with modern number theory assumptions.

Findings

Over rings of stable rank 1, G=UU^-UU^- with length 4.

Under GRH, G over certain rings admits a length 6 factorization.

Without GRH, the length bound is 9 for Hasse domains.

Abstract

Lately, the following problem has attracted a lot of attention in various contexts: find the shortest factorisation $G=UU^-UU^-...U^{\pm}$ of a Chevalley group $G=G(\Phi,R)$ in terms of the unipotent radical $U=U(\Phi,R)$ of the standard Borel subgroup $B=B(\Phi,R)$ and the unipotent radical $U^-=U^-(\Phi,R)$ of the opposite Borel subgroup $B^-=B^-(\Phi,R)$. So far, the record over a finite field was established in a 2010 paper by Babai, Nikolov, and Pyber, where they prove that a group of Lie type admits unitriangular factorisation $G=UU^-UU^-U$ of length 5. Their proof invokes deep analytic and combinatorial tools. In the present paper we notice that from the work of Bass and Tavgen one immediately gets a much more general result, asserting that over any ring of stable rank 1 one has unitriangular factorisation $G=UU^-UU^-$ of length 4. Moreover, we give a detailed survey of triangular factorisations, prove some related results, discuss prospects of generalisation to other classes of rings, and state several unsolved problems. Another main result of the present paper asserts that, in the assumption of the Generalised Riemann's Hypothesis, Chevalley groups over the ring $\Int\Big[\displaystyle{1\over p}\Big]$ admit unitriangular factorisation $G=UU^-UU^-UU^-$ of length 6. Otherwise, the best length estimate for Hasse domains with infinite multiplicative groups that follows from the work of Cooke and Weinberger, gives 9 factors.