Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after
Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
, is a basic result in the branch of
mathematics known as
character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information ab ...
, within
representation theory of a finite group
The representation theory of groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are ...
.
Background
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 theorem (with integer combinations of characters induced from linear characters) could be proved with a collection of subgroups smaller than the Brauer elementary subgroups.
Another result between Artin's induction theorem and Brauer's induction theorem, also due to Brauer and also known as ''Brauer's theorem'' or ''Brauer's lemma'' is the fact that the regular representation of ''G'' can be written as
where the
are ''positive rationals'' and the
are induced from characters of cyclic subgroups of ''G''. Note that in Artin's theorem the characters are induced from the trivial character of the cyclic group, while here they are induced from arbitrary characters (in applications to Artin's ''L'' functions it is important that the groups are cyclic and hence all characters are linear giving that the corresponding ''L'' functions are analytic).
[Serge Lang, ''Algebraic Number Theory'', appendix to chapter XVI]
Statement
Let ''G'' be a
finite group and let Char(''G'') denote the subring of the ring of complex-valued
class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjuga ...
s 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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
) by
induced character In mathematics, an induced character is the character of the representation ''V'' of a finite group ''G'' induced from a representation ''W'' of a subgroup ''H'' ≤ ''G''. More generally, there is also a notion of induction \operatorname(f) of ...
s of the form
, where ''H'' ranges over
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s 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.
Proofs
The proof of Brauer's induction theorem exploits the ring structure of Char(''G'') (most proofs also make use of a slightly larger ring, Char*(G), which consists of