Groups with root-system of type \(BC_\ell\)
In this paper, the classification of groups with a Steinberg presentation (different from type \(G_2\) and \(^2F_4\)) is more or less completed by treating the case of type \(BC_\ell\) (there remain types \(B_\ell\) and \(F_4\) in characteristic 2). The other cases were treated, also quite recently, by the same author in different papers. The case considered in the paper under review is the only one where the root system is non-reduced, i.e., there are roots for which also a nontrivial positive multiple is a root. Roughly speaking, the author proves his result in two steps: the first step is \(\ell=2\), the second \(\ell>2\). In Step 1, he shows that a group with a Steinberg presentation of type \(BC_2\) either has a proper central factor (a kind of degenerate case), or the group is of type \(C_2\), or in the Steinberg relations, equality holds (meaning, for root groups \(A_r\), \(A_s\) with \(r\neq-s\), the commutator \([A_r,A_s]\) is equal to the group generated by all \(A_\lambda\) such that \(\lambda\) is an integer linear combination of \(r\) and \(s\) with strictly positive coefficients). And these cases were considered before by the author. In Step 2, when not in the ``degenerate cases'', the author can use the result of Step 1. Of course, a lot of nontrivial clever arguments are needed to make the proofs complete.
scientific article; zbMATH DE number 1894855