Unitriangular factorisations of Chevalley groups
N.A.Vavilov, A.V.Smolensky, B.Sury

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 of a Chevalley group in terms of the unipotent radical of the standard Borel subgroup and the unipotent radical of the opposite Borel subgroup . 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 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 of length 4. Moreover, we give a detailed survey of…
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Taxonomy
TopicsFinite Group Theory Research · Advanced Algebra and Geometry · Advanced Topics in Algebra
