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 German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation t ...
, is a basic result in the branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
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 a ...
, which is part of the
representation theory of finite groups
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).
Statement
Let ''G'' be a
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
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 conjugati ...
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 commu ...
) by
characters of representations induced from one-dimensional representations of subgroups of ''G''.
In fact, Brauer showed that these subgroups could be chosen from a very restricted collection, now called
elementary subgroups. By definition, a subgroup is elementary if it is a direct product of a cyclic group and a group 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