Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-06-08T20:43:26.832Z Has data issue: false hasContentIssue false
  • English
  • Français

Products of Positive Reflections in the Orthogonal Group

Published online by Cambridge University Press:  20 November 2018

J. Malzan
J. Malzan*
Affiliation:
University of Toronto, Toronto, Ontario
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.

For G a group, S a subset of G which generates G, the length problem in G with respect to S is to find, for g ∈ G, the least integer r such that g can be written as the product of r elements of S. For G an orthogonal group Of(F) (here F is a field, and the elements of Of(F) preserve the quadratic form f) and S the set of reflections in Of(F) the length problem has been studied by E. Cartan [2], J. Dieudonné [4, 5], E. Ellers [7], P. Scherk [8], and others. In all of these investigations, however, the problem posed by requiring that S be a single conjugacy class of reflections in Of(F) has been ignored. And it is generally the case that the reflections in Of(F) fall into several conjugacy classes.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982 , pp. 484 - 499
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Bourgoyne, N. and Cushman, R., Conjugacy classes in linear groups, J. Algebra 44 (1977), 339–362.Google Scholar
2. Cartan, E., Leçons sur la théorie des spineurs (Hermann, Paris, 1938).Google Scholar
3. Chevalley, C., The algebraic theory of spinors (Columbia University Press, New York, 1954), 19–21.Google Scholar
4. J., Dieudonné, Sur les groupes classiques (Hermann, Paris, 1948).Google Scholar
5. J., Dieudonné, La géométrie des groupes classiques (Springer, Berlin-Gôttingen-Heidelberg, 1955), 56–62.Google Scholar
6. Djokovic, D. Z. and Malzan, J., Products of reflections in U(p, q), Memoirs A.M.S., to appear.Google Scholar
7. Ellers, E. W., Decomposition of orthogonal, symplectic, and unitary isometries into simple isometries, Abh. Math. Sem. Univ. Hamburg ^ (1977), 97-127.Google Scholar
8. Scherk, P., On the decomposition of orthogonalities into symmetries, Proc. Amer. Math. Soc. (1950), 481–491.Google Scholar