On Brauer's induction formula of characters of groups

Abstract

and with 2~ linear characters of suitable subgroups of G. Here we will show that for any non-principal irreducible character Z a summation 1,) can be found in which none of the 2] contains the principal character 1 a as an irreducible constituent. This answers in the affirmative a like question of G. O. Michler (Essen, Germany), as put to the author in June 1993. As Michler told him, he could not gather an immediate and positive answer to his query from the existing literature, an observation that the author is willing to share with him after loose inspection of known sources. Note, that a positive answer to Michler's question has consequences for positions of occurrency of zeroes of Dedekind ~-functions and of Artin L-functions of algebraic number fields. To be more specific, let E/K be a finite galois extension of algebraic number fields with gatois group G. Consider the Artin L-function L~: = L(s, 2~, E/K). Then if n~ is the order of the zero or pole of L~ at the point s 0, Artin's conjecture maintains that for )~ ~e 1~ ~ a~n~ < 0 holds whenever Z = Z al 2~, a t ~ ~g, is a (*)-decomposition, in which none of the 2~ contains 1 a as irreducible constituent; that is, L(s, Z, E/K) should be analytic at so. For more information about these topics, one may consult ([2], Chapter VIII), ([3]~ Chapter I) and more in particular the references drawn up in [7]; for very recent developments in respect to the Aramata-Brauer theorem one is referred to [6].