A generalization of the Aramata-Brauer theorem

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.