Decomposition of transvections for automorphisms.
type: Item
instance of: scholarly article
title: Decomposition of transvections for automorphisms.
zbMATH Open document ID: 1211.20043
author: V. G. Kazakevich
author: Nikolai Vavilov
published in: Journal of Mathematical Sciences (New York)
publication date: 2010-01-19T00:00:00Z
review text: Let \(R\) be a commutative ring and \(n\geq 4\). It is known that all automorphisms of \(\text{GL}(n,R)\) are `standard'. Let \(P_i\) denote the maximal parabolic subgroup consisting of the \(g\) that have zeroes in the last \(n-i\) rows of the submatrix formed by the first \(i\) columns. We say that a parabolic subgroup is of type \(i\) if it is conjugate to \(P_i\) under a permutation matrix. Let \(\varphi\) be an automorphism of \(\text{GL}(n,R)\). The last theorem in the paper states that the elementary subgroup \(E(n,R)\) is generated by transvections \(g\) such that \(g\) itself lies in a parabolic of type \(1\), whereas \(\varphi(g)\) lies in the intersection of two parabolics, one of type \(1\) the other of type \(n-1\). This theorem easily implies that \(\varphi\) is `standard', but that was already used in the proof. So it is desirable to find a different argument. The paper contains several similar results and discusses the connection with some basic problems about \(\text{GL}(n,R)\) and \(E(n,R)\).
reviewed by: Wilberd van der Kallen
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 5660144
zbMATH Keywords: general linear groups
zbMATH Keywords: standard automorphisms
zbMATH Keywords: transvections
zbMATH Keywords: elementary matrices
zbMATH Keywords: parabolic subgroups
MaRDI profile type: Publication
full work available at URL: s10958-009-9578-9
OpenAlex ID: W2001794919
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DOI: 10.1007/S10958-009-9578-9
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description: scientific article; zbMATH DE number 5660144