Products of simple isometries of given conjugacy types

Products of simple isometries of given conjugacy types Abstract. Let (V,f) be a finite-dimensional regul r orthogonal vector space over a field .£ and charK 2. Then every e O(K/) is a product = · . . . · ak of at most dim V(n -- 1) + 2 symmetries. We find handsome conditions such that additionally the conjugacy classes can be imposed on k -- l of these symmetries. Analogue theorems are proved for products of transvections in symplectic groups. The special case when K is a euclidean field is studied in greater detail. 1991 Mathematics Subject Classification: 51F25, 15A23, 14L35. 1. Introduction 1.1 Basic assumptions and preliminaries Let K be a field of characteristic distinct from 2 and V a -vector space. All vector spaces will be finite-dimensional. Let/: V x V -> K be a regul r bilinear form such that the assigned orthogonality relation on Fis Symmetrie; hence/is Symmetrie or alternating. Correspondingly, the group of isometries G*= {neGL(V)\f(an,bn) = f(a, b) for all a.beV] is called an orthogonal group O (V,f) or a symplectic group For a subspace Woi Flet rad(J^/) := {w e W\f(v, w) = 0 for every v e W} denote the radical of W. Then/induces a regul r http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Products of simple isometries of given conjugacy types

Forum Mathematicum, Volume 5 (5) – Jan 1, 1993

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1993.5.441
Publisher site
See Article on Publisher Site

Abstract

Abstract. Let (V,f) be a finite-dimensional regul r orthogonal vector space over a field .£ and charK 2. Then every e O(K/) is a product = · . . . · ak of at most dim V(n -- 1) + 2 symmetries. We find handsome conditions such that additionally the conjugacy classes can be imposed on k -- l of these symmetries. Analogue theorems are proved for products of transvections in symplectic groups. The special case when K is a euclidean field is studied in greater detail. 1991 Mathematics Subject Classification: 51F25, 15A23, 14L35. 1. Introduction 1.1 Basic assumptions and preliminaries Let K be a field of characteristic distinct from 2 and V a -vector space. All vector spaces will be finite-dimensional. Let/: V x V -> K be a regul r bilinear form such that the assigned orthogonality relation on Fis Symmetrie; hence/is Symmetrie or alternating. Correspondingly, the group of isometries G*= {neGL(V)\f(an,bn) = f(a, b) for all a.beV] is called an orthogonal group O (V,f) or a symplectic group For a subspace Woi Flet rad(J^/) := {w e W\f(v, w) = 0 for every v e W} denote the radical of W. Then/induces a regul r

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 1993

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