# 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].

How to Cite This Entry:
Brauer characterization of characters. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer_characterization_of_characters&oldid=53322
This article was adapted from an original article by M.E. Harris (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article