Generation of relative commutator subgroups in Chevalley groups

TL;DR

This paper describes generators for the relative commutator subgroups in Chevalley groups over rings, extending classical results and providing explicit generating sets for these subgroups as normal subgroups and groups.

Contribution

It introduces explicit generators for relative commutator subgroups in Chevalley groups, generalizing classical results and applying to a broad class of rings and root systems.

Findings

Generators for the relative elementary subgroup as a normal subgroup.

Explicit generators for the relative commutator subgroup.

Application to relative commutator width in Chevalley groups.

Abstract

Let $\Phi$ be a reduced irreducible root system of rank $\ge 2$, 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 $\big[E(\Phi,R,I),E(\Phi,R,J)\big]$ both as normal subgroups of the elementary Chevalley group $E(\Phi,R)$, and as groups. Namely, let $x_{\a}(\xi)$, $\a\in\Phi$, $\xi\in R$, be an elementary generator of $E(\Phi,R)$. As a normal subgroup of the absolute elementary group $E(\Phi,R)$, the relative elementary subgroup is generated by $x_{\a}(\xi)$, $\a\in\Phi$, $\xi\in I$. Classical results due to Michael Stein, Jacques Tits and Leonid Vaserstein assert that as a group $E(\Phi,R,I)$ is generated by $z_{\a}(\xi,\eta)$, where $\a\in\Phi$, $\xi\in I$, $\eta\in R$. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of $E(\Phi,R)$ the relative commutator subgroup $\big[E(\Phi,R,I),E(\Phi,R,J)\big]$ is generated by the following three types of generators: i) $\big[x_{\alpha}(\xi),z_{\alpha}(\zeta,\eta)\big]$, ii) $\big[x_{\alpha}(\xi),x_{-\alpha}(\zeta)\big]$, and iii) $x_{\alpha}(\xi\zeta)$, where $\alpha\in\Phi$, $\xi\in I$, $\zeta\in J$, $\eta\in R$. As a group, the generators are essentially the same, only that type iii) should be enlarged to iv) $z_{\alpha}(\xi\zeta,\eta)$. For classical groups, these results, with much more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results, namely in the recent work of Alexei Stepanov on relative commutator width.