Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T06:55:31.443Z Has data issue: false hasContentIssue false
  • English
  • Français

Vertex Subgroups of Irreducible Representations of Solvable Groups

Published online by Cambridge University Press:  20 November 2018

J. Malzan
J. Malzan*
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.

If ρ(G) is a finite, real, orthogonal group of matrices acting on the real vector space V, then there is defined [5], by the action of ρ(G), a convex subset of the unit sphere in V called a fundamental region. When the unit sphere is covered by the images under ρ(G) of a fundamental region, we obtain a semi-regular figure.

The group-theoretical problem in this kind of geometry is to find when the fundamental region is unique. In this paper we examine the subgroups, ρ(H), of ρ(G) with a view of finding what subspace, W of V consists of vectors held fixed by all the matrices of ρ(H). Any such subspace lies between two copies of a fundamental region and so contributes to a boundary of both. If enough of these boundaries might be found, the fundamental region would be completely described.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 12 - 21
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Burnside, W. S., Theory of groups of a finite order, 2nd ed. (Dover, New York, 1955).Google Scholar
2. Feit, W., Characters of finite groups (Benjamin, New York-Amsterdam, 1967).Google Scholar
3. Gorenstein, D., Finite groups (Harper and Row, New York-London, 1968).Google Scholar
4. Malzan, J., Real finite linear groups, Ph.D. Thesis, University of Toronto, Toronto, Ontario, 1969.Google Scholar
5. Robinson, G. de B., Geometry of group representations, Nagoya Math. J. 27 (1966), 509–513.Google Scholar
6. Solomon, L., The representation of finite groups in algebraic number fields, J. Math. Soc. Japan 13 (1961), 144–164.Google Scholar