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Conjugacy classes of double covers of monomial groups

Published online by Cambridge University Press:  24 October 2008

John F. Humphreys
John F. Humphreys
Affiliation:
Department of Pure Mathematics, University of Liverpool
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Let G be a finite group, Sn be the symmetric group on n symbols and An be the corresponding alternating group. The conjugacy classes of the wreath product GSn (or monomial group as it is sometimes known) and the conjugacy classes of GAn have been described by Kerber (see [2] and [3]). The group Sn has a double cover n so that the faithful complex representations of this double cover may be regarded as protective representations of Sn. In Section 2, a particular double cover for GSn is constructed, the faithful complex representations of this group being the subject of a joint article with Peter Hoffman[1]. In the present paper, our task is to determine whether a conjugacy class of GSn corresponds to one or to two conjugacy classes in the double cover of GSn (and similarly for GAn). The main results, Theorems 1 and 2, are stated precisely in Section 2 and proved in Sections 3 and 4 respectively. The case when G = 1 provides classical results of Schur ([5], Satz IV). When G is a cyclic group, Read [4] has determined the conjugacy classes, not just for our particular double cover, but for all possible double covers of GSn.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 2 , September 1984 , pp. 195 - 201
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Hoffman, P. and Humphreys, J. Hopf algebras and projective representations of G S n and G A n (to appear).Google Scholar
[2]Kerber, A.. Representations of Permutation Groups. I. Lecture Notes in Math., vol. 240 (Springer-Verlag, 1971).CrossRefGoogle Scholar
[3]Kerber, A.. Representations of Permutation Groups. II. Lecture Notes in Math., vol. 495 (Springer-Verlag, 1975).CrossRefGoogle Scholar
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