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Generation of relative commutator subgroups in Chevalley groups

Published online by Cambridge University Press:  27 October 2015

R. Hazrat ,
N. Vavilov  and
Z. Zhang
R. Hazrat
Affiliation:
Centre for Research in Mathematics, University of Western Sydney, Australia (r.hazrat@uws.edu.au)
N. Vavilov
Affiliation:
Department of Mathematics and Mechanics, Saint Petersburg State University, St Petersburg, Russia (nikolai-vavilov@yandex.ru)
Z. Zhang
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing, People’s Republic of China (zuhong@gmail.com)
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Abstract

Let Φ be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let xα(ξ), α ∈ Φ ξ ∈ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by xα(ξ), α ∈ Φ, ξ ∈ I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Φ,R,I) is generated by zα(ξ,η), where α ∈ Φ, ξ ∈ I, η ∈ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: (i) [xα(ξ),zα(ζ,η)], (ii) [xα(ξ),x_α(ζ)] and (iii) xα(ξζ), where α ∈ Φ, ξ ∈ I, ζ ∈ J, η ∈ R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) zα(ξζ,η). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.

Keywords

elementary generatorsChevalley groupscommutator subgroupsPrimary 20G35
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 393 - 410
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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