Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T08:30:58.248Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological Reflection Groups

Published online by Cambridge University Press:  20 November 2018

Dragomir Ž. Djoković
Dragomir Ž. Djoković*
Affiliation:
University of Waterloo, Waterloo, 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.

Let G be a closed subgroup of one of the classical compact groups 0(n), U(n), Sp(n). By a reflection we mean a matrix in one of these groups which is conjugate to the diagonal matrix diag (–1, 1, …, 1). We say that G is a topological reflection group (t.r.g.) if the subgroup of G generated by all reflections in G is dense in G.

It was shown recently by Eaton and Perlman [5] that, in case of 0(n), the whole group 0(n) is the unique infinite irreducible t.r.g. In this paper we solve the analogous problem for U(n) and Spin). Our method of proof is quite different from the one used in [5]. We treat simultaneously all the three cases.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 294 - 309
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Abe, K., Application of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions, Tôhoku Math. J. 25 (1973), 425–444.Google Scholar
2. Cohen, A. M., Finite complex reflection groups, Ann. Scient, Ecole Norm. Sup. 9 (1970), 379–436.Google Scholar
3. Coxeter, H. S. M., Discrete groups generated by reflections, Annals of Math. 35 (1934), 588–621. Groups generated by unitary reflections of period two, Can. J. Math. 5 (1957), 243-272.Google Scholar
4. Djokovic, D. Z. and Malzan, J., Products of reflections in the quaternionic unitary group, J. of Algebr. 59 (1979), 399–411.Google Scholar
5. Eaton, M. L. and Perlman, M., Generating 0(n) with reflections, Pacific J. Math. 73 (1977), 73–80.Google Scholar
6. Eaton, M. L. and Perlman, M., Reflection groups, generalized Schur functions and the geometry of majorization, Annals of Probabilit. 5 (1977), 829–860.Google Scholar
7. Helgason, S., Differential geometry and symmetric spaces (Academic Press, New York, 1969).Google Scholar
8. Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Vol. 77 (Interscience, New York, 1969).Google Scholar
9. Shephard, G. C. and Todd, J. A., Finite unitary reflection groups, Can. J. Math. 6 (1954), 274–304.Google Scholar
10. Wolf, J. A., Spaces of constant curvature (McGraw-Hill, New York, 1967).Google Scholar