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Linear Groups Generated by Reflection Tori

Published:1999-12-01
Printed: Dec 1999
• A. M. Cohen
• H. Cuypers
• H. Sterk
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Abstract

A reflection is an invertible linear transformation of a vector space fixing a given hyperplane, its axis, vectorwise and a given complement to this hyperplane, its center, setwise. A reflection torus is a one-dimensional group generated by all reflections with fixed axis and center. In this paper we classify subgroups of general linear groups (in arbitrary dimension and defined over arbitrary fields) generated by reflection tori.
 MSC Classifications: 20Hxx - Other groups of matrices [See also 15A30] 20Gxx - Linear algebraic groups and related topics {For arithmetic theory, see 11E57, 11H56; for geometric theory, see 14Lxx, 22Exx; for other methods in representation theory, see 15A30, 22E45, 22E46, 22E47, 22E50, 22E55} 51A50 - Polar geometry, symplectic spaces, orthogonal spaces