Generation of relative commutator subgroups in Chevalley groups. (Q2806123)

Let \(\Phi\) be a reduced irreducible root system of rank greater than or equal to \(2\), let \(R\) be a commutative ring with \(1\) and let \(G(\Phi,R)\) be a Chevalley group of type \(\Phi\) over \(R\). Let \(x_\alpha(\xi)\) be a root unipotent elementary with respect to a fixed maximal split torus in \(G(\Phi,R)\), where \(\alpha\in\Phi\), \(\xi\in R\). The \textit{elementary subgroup} of \(G(\Phi,R)\), \(E(\Phi,R)\), is that generated by all \(x_\alpha(\xi)\). Finally, for each \(R\)-ideal \(I\) the relative elementary subgroup \(E(\Phi,R,I)\) is defined to be the \textit{normal} subgroup of \(E(\Phi,R)\) generated by all \(x_\alpha(\xi)\), where \(\xi\in I\).NEWLINENEWLINE This paper is concerned with the mixed commutator subgroup NEWLINE\[NEWLINEM=[E(\Phi,R,I),E(\Phi,R,J)].NEWLINE\]NEWLINE By means of extensive computations the authors are able (subject to minor restrictions on \(\Phi\)) to provide two simple sets of generators \(S_1,S_2\) for \(M\). \(S_1\) is a generating set for \(M\) as a \textit{normal} subgroup of \(E(\Phi,R)\) while \(S_2\) is a set of generators of \(M\) as a group. Surprisingly \(S_1\) and \(S_2\) differ only slightly. These extend many previous results including some of the authors for classical groups.NEWLINENEWLINE Recently Stepanov has used these results to prove the following remarkable result. Every commutator \([x,y]\), where \(x\in G(\Phi,R,I)\) and \(y\in E(\Phi,R,J)\) is the product of at most \(N\) elements in \(S_2\), where the constant \(N=N(\Phi)\) depends \textit{only} on \(\Phi\) (and is \textit{independent} of the choice of \(R\) and any of its ideals \(I,J\)). (\(G(\Phi,R,I)\) is the \textit{congruence subgroup} of level \(I\) of \(G(\Phi,R)\).)