Characterization of bilinear spaces with unimodular isometry group

Đoković, Dragomir; Szechtman, Fernando

doi:10.1090/S0002-9939-05-07900-1

Characterization of bilinear spaces with unimodular isometry group
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by Dragomir Ž. Đoković and Fernando Szechtman PDF

Proc. Amer. Math. Soc. 133 (2005), 2853-2863 Request permission

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.

References

C. S. Ballantine and E. L. Yip, Congruence and conjunctivity of matrices, Linear Algebra Appl. 32 (1980), 159–198. MR 577914, DOI 10.1016/0024-3795(80)90015-4
B. Corbas and G. D. Williams, Congruence classes in $M_3(\mathbf F_q)$ ($q$ even), Discrete Math. 257 (2002), no. 1, 15–27. MR 1931489, DOI 10.1016/S0012-365X(02)00255-8
B. Corbas and G. D. Williams, Congruence classes in $M_3(\textbf {F}_q)$ ($q$ odd), Discrete Math. 219 (2000), no. 1-3, 37–47. MR 1761709, DOI 10.1016/S0012-365X(99)00366-0
B. Corbas and G. D. Williams, Bilinear forms over an algebraically closed field, J. Pure Appl. Algebra 165 (2001), no. 3, 255–266. MR 1864472, DOI 10.1016/S0022-4049(01)00123-2
Jean Dieudonné, Sur les groupes classiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, VI, Hermann, Paris, 1973 (French). Troisième édition revue et corrigée. MR 0344355
Jean A. Dieudonné, La géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5, Springer-Verlag, Berlin-New York, 1971 (French). Troisième édition. MR 0310083
Dragomir Ž. Đoković and Fernando Szechtman, An elementary proof of Gabriel’s theorem on degenerate bilinear forms and its generalization, J. Algebra 279 (2004), no. 1, 121–125. MR 2078389, DOI 10.1016/j.jalgebra.2004.05.014
Peter Gabriel, Appendix: degenerate bilinear forms, J. Algebra 31 (1974), 67–72. MR 347868, DOI 10.1016/0021-8693(74)90005-2
R. Gow, The number of equivalence classes of nondegenerate bilinear and sesquilinear forms over a finite field, Linear Algebra Appl. 41 (1981), 175–181. MR 649725, DOI 10.1016/0024-3795(81)90097-5
Roderick Gow, The equivalence of an invertible matrix to its transpose, Linear and Multilinear Algebra 8 (1979/80), no. 4, 329–336. MR 572169, DOI 10.1080/03081088008817337
Alexander J. Hahn and O. Timothy O’Meara, The classical groups and $K$-theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989. With a foreword by J. Dieudonné. MR 1007302, DOI 10.1007/978-3-662-13152-7
Jeffrey M. Lee and David A. Weinberg, A note on canonical forms for matrix congruence, Linear Algebra Appl. 249 (1996), 207–215. MR 1417418, DOI 10.1016/0024-3795(95)00357-6
John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372, DOI 10.1007/978-3-642-88330-9
Carl Riehm, The equivalence of bilinear forms, J. Algebra 31 (1974), 45–66. MR 347867, DOI 10.1016/0021-8693(74)90004-0
Rudolf Scharlau, Zur Klassifikation von Bilinearformen und von Isometrien über Körpern, Math. Z. 178 (1981), no. 3, 359–373 (German). MR 635205, DOI 10.1007/BF01214873
Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. MR 770063, DOI 10.1007/978-3-642-69971-9
G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austral. Math. Soc. 3 (1963), 1–62. MR 0150210, DOI 10.1017/S1446788700027622
William C. Waterhouse, The number of congruence classes in $M_n(\mathbf F_q)$, Finite Fields Appl. 1 (1995), no. 1, 57–63. MR 1334625, DOI 10.1006/ffta.1995.1004
J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163.
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4