Brauer's theorem on induced characters | Wikipedia audio article

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SUMMARY
=======
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is, in turn, part of the representation theory of a finite group. Let G be a finite group and let Char(G) denote the subring of the ring of complex-valued class functions of G consisting of integer combinations of irreducible characters. Char(G) is known as the character ring of G, and its elements are known as virtual characters (alternatively, as generalized characters, or sometimes difference characters). It is a ring by virtue of the fact that the product of characters of G is again a character of G. Its multiplication is given by the elementwise product of class functions.
Brauer's induction theorem shows that the character ring can be generated (as an abelian group) by induced characters of the form




λ

H


G




{\displaystyle \lambda _{H}^{G}}
, where H ranges over subgroups of G and λ ranges over linear characters (having degree 1) of H.
In fact, Brauer showed that the subgroups H could be chosen from a very
restricted collection, now called Brauer elementary
subgroups. These are direct products of cyclic groups and groups whose order is a power of a prime.
Using Frobenius reciprocity, Brauer's induction theorem leads easily to his fundamental characterization of characters, which asserts that a complex-valued class function of G is a virtual character if and only if its restriction to each Brauer elementary subgroup of G is a virtual character. This result, together with the fact that a virtual character θ is an irreducible character
if and only if θ(1) 0 and



⟨
θ
,
θ
⟩
=
1


{\displaystyle \langle \theta ,\theta \rangle =1}
(where



⟨
,
⟩


{\displaystyle \langle ,\rangle }
is the usual inner product on the ring of complex-valued class functions) gives
a means of constructing irreducible characters without explicitly constructing the associated representations.
An initial motivation for Brauer's induction theorem was application to Artin L-functions. It shows that those are built up from Dirichlet L-functions, or more general Hecke L-functions. Highly significant for that application is whether each character of G is a non-negative integer combination of characters induced from linear characters of subgroups. In general, this is not the case. In fact, by a theorem of Taketa, if all characters of G are so expressible, then G must be a solvable group (although solvability alone does not guarantee such expressions- for example, the solvable group SL(2,3) has an irreducible complex character of degree 2 which is not expressible as a non-negative integer combination of characters induced from linear characters of subgroups). An ingredient of the proof of Brauer's induction theorem is that when G is a finite nilpotent group, every complex irreducible character of G is induced from a linear character of some subgroup.
A precursor to Brauer's induction theorem was Artin's induction theorem, which states that |G| times the trivial character of G is an integer combination of characters which are each induced from trivial characters of cyclic subgroups of G. Brauer's theorem removes the factor |G|,
but at the expense of expanding the collection of subgroups used. Some years after the proof of Brauer's theorem appeared, J.A. Green showed (in 1955) that no such induction theore ...