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Characterization of bilinear spaces with unimodular isometry group

Authors: Dragomir Z. Ðokovic and Fernando Szechtman
Journal: Proc. Amer. Math. Soc. 133 (2005), 2853-2863
MSC (2000): Primary 15A63
Published electronically: March 29, 2005
MathSciNet review: 2159762
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Abstract: We study finite-dimensional bilinear spaces and their isometry groups. To each bilinear space $V$ we associate two canonical filtrations, which yield structural results on $V$. Prominent among these is an explicit formula for the number of indecomposable and degenerate blocks of $V$ of a given dimension. Equipped with this material, we proceed to characterize those bilinear spaces whose isometry group is contained in the special linear group. This characterization can easily be implemented in practice by means of an algorithm. As an application, we determine the real $n$-by-$n$ matrices whose congruence class is disconnected.

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Additional Information

Dragomir Z. Ðokovic
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Fernando Szechtman
Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2

Received by editor(s): November 7, 2003
Received by editor(s) in revised form: May 27, 2004
Published electronically: March 29, 2005
Additional Notes: The first author was supported in part by the NSERC Grant A-5285.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2005 American Mathematical Society

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