St. Petersburg State University, Russia
January 27, 2011
New results on subgroups of classical groups: We give an account of some recent results on description of subgroups of a [classical] Chevalley group $G(\Phi,R)$ of type $\Phi$ over a commutative ring $R$, containing an elementary subgroup $\phi(E(\Delta,A))$ in a rational representation $\phi$.
A natural context to specify broad classes of large semi-simple subgroups in classical groups is provided by Aschbacher's subgroup structure theorem, and its generalisation to exceptional groups by Liebeck and Seitz.
Until recently, little was known on description of subgroups from Aschbacher classes. The only case which was completely settled in the 1980-ies, originally by Borewicz and the author, were overgroups of subsystem subgroups, class $C_1+C_2$. Generalisations of these results to other classes were widely discussed, but no definitive results were in sight until 2000.
Over the last decade the situation changed dramatically. Petrov, the author, You Hong, Luzgarev, Stepanov, Ananievsky, Sinchuk succeeded in completely settling the problem of standard description for overgroups of classical subgroups and subring subgroups, classes $C_8$ and $C_5$, and made significant progress towards description of overgroups for tensored subgroups, class $C_4$.
The common feature of all these results is that they turned out to be technically terribly much more demanding than what was expected in the 1990-ies. Oftentimes, even the answers themselves are very different from the conjectured ones. Here, one should from the very start take account of the effects from the theory of algebraic groups, whereas the proofs heavily rely on the power of localisation methods, such as Quillen--Suslin--Vaserstein localisation and patching, or Bak's localisation-completion.