Brauer's theorem on induced characters
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In mathematics, Brauer's theorem on induced characters is a basic result in the representation theory of a finite group. Let G be a finite group and Char(G) the abelian group of virtual characters of G; in other words, take the traces of the irreducible representations of G, in the sense of linear representations over the complex numbers, and look at the abelian group of complex-valued functions on G that these generate. These are all class functions, and there arises the question of saying in some other way which class functions they are.
Richard Brauer's result states that Char(G) can also be generated as abelian group by its subset of characters of the induced representations
- IndHG(ρ)
where H is any subgroup of G and ρ a degree one character (in other words a group homomorphism from H to the circle group).
In simpler language, the characters of G are linear combinations with integer coefficients of characters induced from characters of subgroups into the roots of unity. The initial motivation of this was application to Artin L-functions. It shows that in some serious sense those are built up from Dirichlet L-functions, or more general Hecke L-functions. Highly significant for that application is whether the coefficients of the linear combination can be taken non-negative; the answer to that is in general no.
This result was proved in 1946. In 1986 Victor Snaith gave a new kind of proof, topological in nature (application of the Lefschetz fixed-point theorem). This has led to further work on the question of finding natural and explicit versions of this theorem.
See also: Artin's induction theorem.
