Brauer characterization of characters
Let be a field, let
be a finite-dimensional vector space over
and let
be a finite group. A representation of
over
is a group homomorphism
(the group of
-linear automorphisms of
) or, equivalently, a module action of the group algebra
on
(the equivalence is defined by:
for all
; cf. also Representation of a group). The character
is defined by:
for all
. Since
for any two
-matrices
,
over
, one finds that
for all
, and hence
is a class function on
. Equivalent representations have the same character and
is the sum of the characters of the quotient
-modules in any
-filtration of
. If
acts irreducibly on
, then
is said to be an irreducible
-module and
is said to be an irreducible character.
If , then
is said to be a linear representation of
, and
is an irreducible
-module;
is said to be a linear character. There are at most
inequivalent types of irreducible representations of
over
. Let
be the set of irreducible characters of
over
. Then
is
-linearly independent in
and every character of
over
is a sum of elements of
.
Let be a subgroup of
and let
be a representation of
. Clearly, the restriction to
,
, is a representation of
and induction to
,
, is a representation of
(cf. also Induced representation). If
is a transversal for the right cosets of
in
, then, for
,
![]() |
where if
and
if
.
If and
are
-modules, then
is a
-module defined by "diagonal action" :
![]() |
for all ,
and
, and
.
Assume that , the field of complex numbers. Then
is a
-basis of
. Also, every
-module
is completely reducible (i.e., a direct sum of irreducible submodules). Also,
,
(complex conjugate) and
is a sum of
-th roots of unity for all
. Also, there is a non-singular symmetric scalar product
defined by:
![]() |
Here, (the Kronecker delta) for all
and if
and
are two finite-dimensional
-modules, then
and hence the isomorphism type of
is determined by
.
If is a prime integer, then a finite group that is the direct product of a cyclic group and a
-group (or equivalently of a cyclic
-group and a
-group) is called a
-elementary group. Any subgroup or quotient of such a group is also
-elementary. A finite group is called elementary if it is
-elementary for some prime number
. It is well-known that each irreducible character
of an elementary group
is of the form
for some subgroup
of
and some linear character
of
(cf. [a8], Thm. 16).
For a finite group , let
be the additive subgroup of
generated by all characters of
. The elements of
are called virtual or generalized characters of
and
is a ring and also a free Abelian group with free basis
. Clearly,
![]() |
In [a2], R. Brauer proved the following assertions:
1) Every character of a finite group
is a linear combination with integer coefficients of characters induced from linear characters of elementary subgroups of
.
Brauer used this result in [a2] to prove that Artin -functions of virtual characters have a meromorphic extension to the entire complex plane. Then, in [a3], he proved that this assertion is equivalent to what is known as the Brauer characterization of characters:
2) A class function lies in
if and only if
for every elementary subgroup
of
.
An immediate consequence (cf. [a8], Thm 22 and Corollary) is:
3) A class function lies in
if and only if for each elementary subgroup
of
and each linear character
of
,
![]() |
A sort of converse of 1) was given by J. Green ([a8], Thm. 23{}). There are numerous applications of these results (cf. [a7], Lemma 8.14; Thm. 8.24, [a6], V, Hauptsatz 19.11, [a8], Sect. 11.2; Chap. 12).
Significant improvements to the proofs of these results have been obtained by several authors [a4], [a7], Chap. 8, [a8], Chaps. 10, 11, [a6], V, Sect. 19.
Let denote the free Abelian group whose free basis is given by the
-conjugacy classes
, where
is a subgroup of
and
is a linear character of
. Clearly
is a character of
and hence induction induces an Abelian group homomorphism
, which is surjective by 1). Some interesting recent results in [a9] and [a1] give explicit (functorial) splittings of
(i.e., an explicit group homomorphism
such that
).
Clearly, is the Grothendieck group of the category of finitely generated
-modules. Consequently, 1) can be viewed as proving the surjectivity of the induction mapping from one Grothendieck group into another. By changing the coefficient ring
, or by considering the modular context, etc., many important analogues of these results emerge, cf. [a8], Chaps. 12, 16, 17, [a5], Thm. 2, [a10].
References
[a1] | R. Boltje, "A canonical Brauer induction formula" Asterisque , 181/2 (1990) pp. 31–59 |
[a2] | R. Brauer, "On Artin's ![]() |
[a3] | R. Brauer, "A characterization of the characters of a group of finite order" Ann. of Math. , 57 (1953) pp. 357–377 |
[a4] | R. Brauer, J. Tate, "On the characters of finite groups" Ann. of Math. , 62 (1955) pp. 1–7 |
[a5] | M. Broué, "Sur l'induction des modules indecomposables et la projectraité relative" Math. Z. , 149 (1976) pp. 227–245 |
[a6] | B. Huppert, "Endliche Gruppen" , I , Springer (1967) pp. Chapt. V |
[a7] | I.M. Isaacs, "Character theory of finite groups" , Acad. Press (1976) |
[a8] | J.-P. Serre, "Linear representations of finite groups" , Springer (1977) (Translated from French) |
[a9] | V. Snaith, "Explicit Brauer induction" Invent. Math. , 94 (1988) pp. 455–478 |
[a10] | X. Zhou, "On the decomposition map of Grothendieck groups" Math. Z. , 206 (1991) pp. 533–534 |
Brauer characterization of characters. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer_characterization_of_characters&oldid=18401