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 -series with general group characters" Ann. of Math. , 48 (1947) pp. 502–514
[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