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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
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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

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