Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T19:25:09.763Z Has data issue: false hasContentIssue false
  • English
  • Français

On the conjugacy classes in the unitary, symplectic and orthogonal groups

Published online by Cambridge University Press:  09 April 2009

G. E. Wall
G. E. Wall
Affiliation:
Department of Mathematics, University of Sydney.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is concerned with two mani problems:

(a) the determination of the conjugacy classes in the finite-dimensional unitary, symplectic and orthogonal groups over division rings or fields;

(b) the determination of the equivalence classes of non-degenerate sesquilinear forms on finite-dimensional vector spaces.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 3 , Issue 1 , February 1963 , pp. 1 - 62
Copyright
Copyright © Australian Mathematical Society 1963

References

[1]Bourbaki, N., Formes sesquilnéaires et formes quadratiques (Éléments de Mathématique I, liver II, chapitre 9), Hermann (Paris), 1959.Google Scholar
[2]Dieudonné, J., La géometrie des groups classiques, Springer (Berlin), 1955.Google Scholar
[3]Feit, W. and Fine, N. J., Paris of commuting matrices over a finite field Duke Math. J. 27 (1960), 91–94.Google Scholar
[4]Ingraham, M. H. and Wegner, K. W., The equivalence of pairs of Hermitian matrices, Prans. Amer. Math. Soc. 38 (1935), 145–162.Google Scholar
[5]Jacobson, N., Lectures in Abstract Algebra, vol. II, Van Nostrand (New York), 1952.Google Scholar
[6]Jacobson, N., The theory of rings, Math. Surveys No. II, Amercian mathematyical Society (New York), 1943.Google Scholar
[7]Klingenberg, W., Paare symmetrischer und Formen zweiten Grades Abh. Math. Sem. Uinv. Hamburg 19 (1954), 78–93.Google Scholar
[8]Pickert, G., Normalformen von Matrizen, Enz. der Math. Wiss. Bd. Il, Heft 3, Teil 1, Teubner (Leipzig), 1953.Google Scholar
[9]Springer, T. A., Over Synplectische Transformaties, Thesis, University of Leiden, 1951.Google Scholar
[10]Trott, G. R., On the canonical form of a non-singular pencil of Hermitian matrices, Amer. J. Math. 56 (1934), 359–371.Google Scholar
[11]Turnbull, H. W., On the equivalence of pencils of Hermitian forms, Proc. London Math. Soc. (2) 39 (1935), 232–248.Google Scholar
[12]Vekatachalienger, K., Pairs of symmetric and skew matrices in an arbitrary field 1, Proc. Indian Acad. Sci., Sect. A, 22 (1945), 243–264.Google Scholar
[13]Wall, G. E., The structure of a unitary factor group, Publ. math. de l 'Inst. des hautes études scientifique No. 1 (1959).Google Scholar
[14]Williamson, J., The equivalence of non-singular pencils of Hermitian matrices in an arbitraty field, Amer. J. Math. 57 (1935), 475–490.Google Scholar
[15]Williamson, J., On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math. 58 (1936), 141–163.Google Scholar
[16]Williamson, J., Quasi-unitary matrices, Duke Math. J. 3 (1937) 715–725.Google Scholar
[17]Williamson, J., On the normal forms of linear canonical transformations in dynamics, Amer. J. Math. 59 (1937), 599–617.Google Scholar
[18]Williamson, J., Normal matrices over an arbitrary field of characteristic zero, Amer. J. Math. 61 (1939), 335–358.Google Scholar
[19]Williamson, J., Note on the equivalence of non-singular pencils of Hermitian matrices, Bull. Amer. Math. Soc. 51 (1945), 894–897.Google Scholar
[20]Zassenhaus, H., On a normal form of the orthogonal transformation, Canad. Math. Bull. 1 (1958), (I) 31–39, (II) 101–111, (III) 183–191.Google Scholar
[21]Ennola, V.On the conjugacy classes of the finite unitary groups, Ann. Acad. Sci. Fenn. AI, No. 313 (1962).Google Scholar