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Rational valued and real valued projective characters of finite groups

Published online by Cambridge University Press:  18 May 2009

J. F. Humphreys
J. F. Humphreys
Affiliation:
The University, Liverpool, England
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It is well-known [3; V.13.7] that each irreducible complex character of a finite group G is rational valued if and only if for each integer m coprime to the order of G and each g ∈ G, g is conjugate to gm. In particular, for each positive integer n, the symmetric group on n symbols, S(n), has all its irreducible characters rational valued. The situation for projective characters is quite different. In [5], Morris gives tables of the spin characters of S(n) for n ≤ 13 as well as general information about the values of these characters for any symmetric group. It can be seen from these results that in no case are all the spin characters of S(n) rational valued and, indeed, for n ≥ 6 these characters are not even all real valued. In section 2 of this note, we obtain a necessary and sufficient condition for each irreducible character of a group G associated with a 2-cocycle α to be rational valued. A corresponding result for real valued projective characters is discussed in section 3. Section 1 contains preliminary definitions and notation, including the definition of projective characters given in [2].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 21 , Issue 1 , January 1980 , pp. 23 - 28
Copyright
Copyright © Glasgow Mathematical Journal Trust 1980

References

REFERENCES

1.DeMeyer, F. R. and Janusz, G. J., Finite groups with an irreducible representation of large degree, Math. Z. 108 (1969), 145–153.CrossRefGoogle Scholar
2.Haggarty, R. J. and Humphreys, J. F., Projective characters of finite groups, Proc. London Math. Soc. (3) 36 (1978), 176–192.CrossRefGoogle Scholar
3.Huppert, B., Endliche Gruppen I. (Springer, 1967).CrossRefGoogle Scholar
4.Isaacs, I. M., Character theory of finite groups, (Academic Press, 1976).Google Scholar
5.Morris, A. O., The spin representation of the symmetric group, Proc. London Math. Soc. (3) 12 (1962), 55–76.CrossRefGoogle Scholar
6.Morris, A. O., Projective characters of exceptional Weyl groups, J. Algebra 29 (1974), 567–586.CrossRefGoogle Scholar