Brauer characterization of characters

Let $ K $ be a field, let $ V $ be a finite-dimensional vector space over $ K $ and let $ G $ be a finite group. A representation of $ G $ over $ V $ is a group homomorphism $ X : G \rightarrow { { \mathop{\rm GL} } ( V/K ) } $( the group of $ K $- linear automorphisms of $ V $) or, equivalently, a module action of the group algebra $ KG $ on $ V $( the equivalence is defined by: $ g \cdot v = X ( g ) v $ for all $ g,v \in V $; cf. also Representation of a group). The character $ {\chi _ {V} = { \mathop{\rm char} } ( X ) } : G \rightarrow K $ is defined by: $ \chi _ {V} ( g ) = { \mathop{\rm tr} } ( X ( g ) ) $ for all $ g \in G $. Since $ { \mathop{\rm tr} } ( AB ) = { \mathop{\rm tr} } ( BA ) $ for any two $ ( n \times n ) $- matrices $ A $, $ B $ over $ K $, one finds that $ \chi _ {V} ( ghg ^ {-1 } ) = \chi ( h ) $ for all $ g,h \in G $, and hence $ \chi _ {V} $ is a class function on $ G: \chi _ {V} \in { \mathop{\rm CF} } ( G,K ) $. Equivalent representations have the same character and $ \chi _ {V} $ is the sum of the characters of the quotient $ KG $- modules in any $ KG $- filtration of $ V $. If $ G $ acts irreducibly on $ V/K $, then $ V $ is said to be an irreducible $ KG $- module and $ \chi _ {V} $ is said to be an irreducible character.

If $ { \mathop{\rm dim} } ( V/K ) = 1 $, then $ V $ is said to be a linear representation of $ G $, and $ V $ is an irreducible $ KG $- module; $ \chi _ {V} $ is said to be a linear character. There are at most $ | G | $ inequivalent types of irreducible representations of $ G $ over $ K $. Let $ { \mathop{\rm Irr} } _ {K} ( G ) = \{ \chi _ {1} \dots \chi _ {k} \} $ be the set of irreducible characters of $ G $ over $ K $. Then $ { \mathop{\rm Irr} } _ {K} ( G ) $ is $ K $- linearly independent in $ { \mathop{\rm CF} } ( G,K ) $ and every character of $ G $ over $ K $ is a sum of elements of $ { \mathop{\rm Irr} } _ {K} ( G ) $.

Let $ H $ be a subgroup of $ G $ and let $ {\mathcal Y} : H \rightarrow { { \mathop{\rm GL} } ( W/K ) } $ be a representation of $ H $. Clearly, the restriction to $ H $, $ { { \mathop{\rm Res} } _ {H} ^ {G} ( X ) } : H \rightarrow { { \mathop{\rm GL} } ( V/K ) } $, is a representation of $ H $ and induction to $ G $, $ { { \mathop{\rm Ind} } _ {H} ^ {G} ( {\mathcal Y} ) } : G \rightarrow { { \mathop{\rm GL} } ( KG \otimes _ {KH } W ) } $, is a representation of $ G $( cf. also Induced representation). If $ T $ is a transversal for the right cosets of $ H $ in $ G $, then, for $ g \in G $,

$$ { \mathop{\rm char} } ( { \mathop{\rm Ind} } _ {H} ^ {G} ( {\mathcal Y} ) ) ( g ) = \sum _ {t \in T } { \mathop{\rm char} } ( {\mathcal Y} ) ^ {o} ( tgt ^ {-1 } ) , $$

where $ { \mathop{\rm char} } ^ {o} ( {\mathcal Y} ) ( u ) = { \mathop{\rm Char} } ( {\mathcal Y} ) ( u ) $ if $ u \in H $ and $ { \mathop{\rm char} } ^ {o} ( {\mathcal Y} ) ( u ) = 0 $ if $ u \in G - H $.

If $ V $ and $ W $ are $ KG $- modules, then $ V \otimes _ {K} W $ is a $ KG $- module defined by "diagonal action" :

$$ g ( v \otimes _ {K} w ) \equiv ( gv ) \otimes _ {K} ( gw ) $$

for all $ v \in V $, $ w \in W $ and $ g \in G $, and $ { \mathop{\rm char} } ( V \otimes _ {K} W ) = { \mathop{\rm char} } ( V ) { \mathop{\rm char} } ( W ) $.

Assume that $ K = \mathbf C $, the field of complex numbers. Then $ { \mathop{\rm Irr} } _ {\mathbf C} ( G ) $ is a $ \mathbf C $- basis of $ { \mathop{\rm CF} } ( G, \mathbf C ) $. Also, every $ \mathbf C G $- module $ V $ is completely reducible (i.e., a direct sum of irreducible submodules). Also, $ \chi _ {V} ( 1 ) = { \mathop{\rm dim} } ( V/ \mathbf C ) $, $ \chi ( g ^ {- 1 } ) = {\overline{ {\chi ( g ) }}\; } $( complex conjugate) and $ \chi ( g ) $ is a sum of $ \chi ( 1 ) $ $ | g | $- th roots of unity for all $ g \in G $. Also, there is a non-singular symmetric scalar product $ {\langle {\cdot, \cdot } \rangle } : { { \mathop{\rm CF} } ( G, \mathbf C ) \times { \mathop{\rm CF} } ( G, \mathbf C ) } \rightarrow \mathbf C $ defined by:

$$ \left \langle {f _ {1} ,f _ {2} } \right \rangle = { \frac{1}{\left | G \right | } } \sum _ {g \in G } f _ {1} ( g ) f _ {2} ( g ^ {-1 } ) . $$

Here, $ \langle {\chi _ {i} , \chi _ {j} } \rangle = \delta _ {ij } $( the Kronecker delta) for all $ \chi _ {i} , \chi _ {j} \in { \mathop{\rm Irr} } _ {\mathbf C} ( G ) $ and if $ V $ and $ W $ are two finite-dimensional $ \mathbf C G $- modules, then $ { \mathop{\rm dim} } ( { \mathop{\rm Hom} } _ {\mathbf C G } ( V,W ) / \mathbf C ) = \langle {\chi _ {V} , \chi _ {W} } \rangle $ and hence the isomorphism type of $ V $ is determined by $ \chi _ {V} $.

If $ p $ is a prime integer, then a finite group that is the direct product of a cyclic group and a $ p $- group (or equivalently of a cyclic $ p ^ \prime $- group and a $ p $- group) is called a $ p $- elementary group. Any subgroup or quotient of such a group is also $ p $- elementary. A finite group is called elementary if it is $ p $- elementary for some prime number $ p $. It is well-known that each irreducible character $ \chi $ of an elementary group $ E $ is of the form $ \chi = { \mathop{\rm Ind} } _ {H} ^ {E} ( \psi ) $ for some subgroup $ H $ of $ E $ and some linear character $ \psi $ of $ H $( cf. [a8], Thm. 16).

For a finite group $ G $, let $ R ( G ) $ be the additive subgroup of $ { \mathop{\rm CF} } ( G, \mathbf C ) $ generated by all characters of $ G $. The elements of $ R ( G ) $ are called virtual or generalized characters of $ G $ and $ R ( G ) $ is a ring and also a free Abelian group with free basis $ { \mathop{\rm Irr} } _ {\mathbf C} ( G ) $. Clearly,

$$ { \mathop{\rm Irr} } _ {\mathbf C} ( G ) = \left \{ {\varphi \in R ( G ) } : {\left \langle {\varphi, \varphi } \right \rangle =1 \textrm{ and } \varphi ( 1 ) > 0 } \right \} . $$

In [a2], R. Brauer proved the following assertions:

1) Every character $ \chi $ of a finite group $ G $ is a linear combination with integer coefficients of characters induced from linear characters of elementary subgroups of $ G $.

Brauer used this result in [a2] to prove that Artin $ L $- 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 $ \varphi \in { \mathop{\rm CF} } ( G, \mathbf C ) $ lies in $ R ( G ) $ if and only if $ { \mathop{\rm Res} } _ {E} ^ {G} ( \varphi ) \in R ( E ) $ for every elementary subgroup $ E $ of $ G $.

An immediate consequence (cf. [a8], Thm 22 and Corollary) is:

3) A class function $ \varphi \in { \mathop{\rm CF} } ( G, \mathbf C ) $ lies in $ R ( G ) $ if and only if for each elementary subgroup $ E $ of $ G $ and each linear character $ \chi $ of $ E $,

$$ \left \langle { { \mathop{\rm Res} } _ {E} ^ {G} ( \varphi ) , \chi } \right \rangle \in Z. $$

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 $ R _ {+} ( G ) $ denote the free Abelian group whose free basis is given by the $ G $- conjugacy classes $ ( H, \lambda ) $, where $ H $ is a subgroup of $ G $ and $ \lambda $ is a linear character of $ H $. Clearly $ { \mathop{\rm Ind} } _ {H} ^ {G} ( \lambda ) $ is a character of $ G $ and hence induction induces an Abelian group homomorphism $ {\mathcal I} : {R _ {+} ( G ) } \rightarrow {R ( G ) } $, which is surjective by 1). Some interesting recent results in [a9] and [a1] give explicit (functorial) splittings of $ {\mathcal I} $( i.e., an explicit group homomorphism $ {\mathcal J} : {R ( G ) } \rightarrow {R _ {+} ( G ) } $ such that $ {\mathcal I} \cdot {\mathcal J} = { \mathop{\rm id} } _ {R ( G ) } $).

Clearly, $ R ( G ) $ is the Grothendieck group of the category of finitely generated $ \mathbf C G $- 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 $ \mathbf C $, 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 L-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 indécomposables et la projectivité 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