Decomposition of transvections for automorphisms

N. A. Vavilov   ; V. G. Kazakevich

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 18, Tome 365 (2009), pp. 47-62

Résumé

The method of decomposition of unipotents consists in writing elementary matrices as products of factors lying in proper parabolic subgroups, whose images under (abstract) inner automorphisms also fall into proper parabolic subgroups of certain types. For the general linear group, this method was first proposed by Stepanov in 1987 to simplify the proof of Suslin's normality theorem. Soon thereafter Vavilov and Plotkin generalised it to other classical groups and Chevalley groups. Subsequently, many further results of that type have been discovered. In the present paper, we show that a similar decomposition can be constructed for arbitrary standard automorphisms. This result emerged in the context of a simplified proof of the theorems due to Waterhouse, Golubchik, Mikhalev, Zelmanov, and Petechuk regarding the standard description of automorphisms of the general linear group, based exclusively on the use of unipotent elements. Bibl. – 27 titles.

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