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A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

Published online by Cambridge University Press:  20 November 2018

Hirosi Nagao
Hirosi Nagao*
Affiliation:
Osaka City University The University of Toronto
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Extract

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Let G be a finite group of order g, and

be an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :

1

It is easy to conclude from (1) the following orthogonality relations for characters:

where

and is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 59 - 60
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Schur, I., Neue Begründung der Theorie der Gruppencharactere, Sitzungsber. Preuss. Akad. d. Wiss. (1905), 406-32.Google Scholar