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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization of the Aramata-Brauer theorem


Author: Sandra L. Rhoades
Journal: Proc. Amer. Math. Soc. 119 (1993), 357-364
MSC: Primary 20C15; Secondary 11M41, 11R42
MathSciNet review: 1166360
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Abstract: The Aramata-Brauer Theorem says that the regular character minus the principal character of a finite group can be written as a positive rational linear combination of induced linear characters. In the language of Artin $ L$-series this says that $ {\zeta _E}(s)/{\zeta _F}(s)$ is entire, where this is the quotient of the Dedekind $ \zeta $-functions of a Galois extension $ E/F$ of number fields. Given any subset of characters of a finite group, we will give a necessary and sufficient condition for when a character is a positive rational linear combination of characters from this specified subset. This result implies that the regular character plus or minus any irreducible character can be written as a positive rational linear combination of induced linear characters. This both generalizes and gives a new proof of the Aramata-Brauer Theorem.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1166360-8
PII: S 0002-9939(1993)1166360-8
Article copyright: © Copyright 1993 American Mathematical Society