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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b1108501.png" /> be a [[Field|field]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b1108502.png" /> be a finite-dimensional [[Vector space|vector space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b1108503.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b1108504.png" /> be a [[Finite group|finite group]]. A representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b1108505.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b1108506.png" /> is a group [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b1108507.png" /> (the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b1108508.png" />-linear automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b1108509.png" />) or, equivalently, a module action of the [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085011.png" /> (the equivalence is defined by: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085013.png" />; cf. also [[Representation of a group|Representation of a group]]). The character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085014.png" /> is defined by: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085016.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085017.png" /> for any two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085018.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085020.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085021.png" />, one finds that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085022.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085023.png" />, and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085024.png" /> is a class function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085025.png" />. Equivalent representations have the same character and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085026.png" /> is the sum of the characters of the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085027.png" />-modules in any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085028.png" />-filtration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085029.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085030.png" /> acts irreducibly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085032.png" /> is said to be an irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085034.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085035.png" /> is said to be an irreducible character.
+
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$#C+1 = 159 : ~/encyclopedia/old_files/data/B110/B.1100850 Brauer characterization of characters
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085036.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085037.png" /> is said to be a linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085038.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085039.png" /> is an irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085040.png" />-module; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085041.png" /> is said to be a linear character. There are at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085042.png" /> inequivalent types of irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085043.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085044.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085045.png" /> be the set of irreducible characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085046.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085047.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085048.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085049.png" />-linearly independent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085050.png" /> and every character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085051.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085052.png" /> is a sum of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085053.png" />.
+
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085054.png" /> be a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085055.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085056.png" /> be a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085057.png" />. Clearly, the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085059.png" />, is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085060.png" /> and induction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085062.png" />, is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085063.png" /> (cf. also [[Induced representation|Induced representation]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085064.png" /> is a transversal for the right cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085065.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085066.png" />, then, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085067.png" />,
+
Let $  K $
 +
be a [[Field|field]], let  $  V $
 +
be a finite-dimensional [[Vector space|vector space]] over  $  K $
 +
and let $  G $
 +
be a [[Finite group|finite group]]. A representation of $  G $
 +
over  $  V $
 +
is a group [[Homomorphism|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|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|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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085068.png" /></td> </tr></table>
+
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 ) $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085069.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085071.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085072.png" />.
+
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|Induced representation]]). If  $  T $
 +
is a transversal for the right cosets of  $  H $
 +
in  $  G $,
 +
then, for  $  g \in G $,
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085074.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085075.png" />-modules, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085076.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085077.png" />-module defined by  "diagonal action" :
+
$$
 +
{ \mathop{\rm char} } ( { \mathop{\rm Ind} } _ {H}  ^ {G} ( {\mathcal Y} ) ) ( g ) = \sum _ {t \in T } { \mathop{\rm char} } ( {\mathcal Y} )  ^ {o} ( tgt ^ {-1 } ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085078.png" /></td> </tr></table>
+
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 $.
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085081.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085082.png" />.
+
If  $  V $
 +
and $  W $
 +
are  $  KG $-
 +
modules, then  $  V \otimes _ {K} W $
 +
is a  $  KG $-
 +
module defined by  "diagonal action" :
  
Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085083.png" />, the field of complex numbers. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085084.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085085.png" />-basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085086.png" />. Also, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085087.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085088.png" /> is completely reducible (i.e., a direct sum of irreducible submodules). Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085090.png" /> (complex conjugate) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085091.png" /> is a sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085092.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085093.png" />-th roots of unity for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085094.png" />. Also, there is a non-singular symmetric scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085095.png" /> defined by:
+
$$
 +
g ( v \otimes _ {K} w ) \equiv ( gv ) \otimes _ {K} ( gw )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085096.png" /></td> </tr></table>
+
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 ) $.
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085097.png" /> (the Kronecker delta) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085098.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b11085099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850100.png" /> are two finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850101.png" />-modules, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850102.png" /> and hence the isomorphism type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850103.png" /> is determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850104.png" />.
+
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:
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850105.png" /> is a prime integer, then a [[Finite group|finite group]] that is the [[Direct product|direct product]] of a [[Cyclic group|cyclic group]] and a [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850106.png" />-group]] (or equivalently of a cyclic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850107.png" />-group and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850108.png" />-group) is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850110.png" />-elementary group. Any subgroup or quotient of such a group is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850111.png" />-elementary. A finite group is called elementary if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850112.png" />-elementary for some prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850113.png" />. It is well-known that each irreducible character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850114.png" /> of an elementary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850115.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850116.png" /> for some subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850117.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850118.png" /> and some linear character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850119.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850120.png" /> (cf. [[#References|[a8]]], Thm. 16).
+
$$
 +
\left \langle  {f _ {1} ,f _ {2} } \right \rangle = {
 +
\frac{1}{\left | G \right | }
 +
} \sum _ {g \in G } f _ {1} ( g ) f _ {2} ( g ^ {-1 } ) .
 +
$$
  
For a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850121.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850122.png" /> be the additive subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850123.png" /> generated by all characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850124.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850125.png" /> are called virtual or generalized characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850127.png" /> is a ring and also a free Abelian group with free basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850128.png" />. Clearly,
+
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} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850129.png" /></td> </tr></table>
+
If  $  p $
 +
is a prime integer, then a [[Finite group|finite group]] that is the [[Direct product|direct product]] of a [[Cyclic group|cyclic group]] and a [[P-group| $  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. [[#References|[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 [[#References|[a2]]], R. Brauer proved the following assertions:
 
In [[#References|[a2]]], R. Brauer proved the following assertions:
  
1) Every character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850130.png" /> of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850131.png" /> is a linear combination with integer coefficients of characters induced from linear characters of elementary subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850132.png" />.
+
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 [[#References|[a2]]] to prove that Artin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850133.png" />-functions of virtual characters have a meromorphic extension to the entire complex plane. Then, in [[#References|[a3]]], he proved that this assertion is equivalent to what is known as the Brauer characterization of characters:
+
Brauer used this result in [[#References|[a2]]] to prove that Artin $  L $-
 +
functions of virtual characters have a meromorphic extension to the entire complex plane. Then, in [[#References|[a3]]], he proved that this assertion is equivalent to what is known as the Brauer characterization of characters:
  
2) A class function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850134.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850135.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850136.png" /> for every elementary subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850137.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850138.png" />.
+
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. [[#References|[a8]]], Thm 22 and Corollary) is:
 
An immediate consequence (cf. [[#References|[a8]]], Thm 22 and Corollary) is:
  
3) A class function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850139.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850140.png" /> if and only if for each elementary subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850141.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850142.png" /> and each linear character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850143.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850144.png" />,
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850145.png" /></td> </tr></table>
+
$$
 +
\left \langle  { { \mathop{\rm Res} } _ {E}  ^ {G} ( \varphi ) , \chi } \right \rangle \in Z.
 +
$$
  
 
A sort of converse of 1) was given by J. Green ([[#References|[a8]]], Thm. 23{'''}). There are numerous applications of these results (cf. [[#References|[a7]]], Lemma 8.14; Thm. 8.24, [[#References|[a6]]], V, Hauptsatz 19.11, [[#References|[a8]]], Sect. 11.2; Chap. 12).
 
A sort of converse of 1) was given by J. Green ([[#References|[a8]]], Thm. 23{'''}). There are numerous applications of these results (cf. [[#References|[a7]]], Lemma 8.14; Thm. 8.24, [[#References|[a6]]], V, Hauptsatz 19.11, [[#References|[a8]]], Sect. 11.2; Chap. 12).
Line 45: Line 199:
 
Significant improvements to the proofs of these results have been obtained by several authors [[#References|[a4]]], [[#References|[a7]]], Chap. 8, [[#References|[a8]]], Chaps. 10, 11, [[#References|[a6]]], V, Sect. 19.
 
Significant improvements to the proofs of these results have been obtained by several authors [[#References|[a4]]], [[#References|[a7]]], Chap. 8, [[#References|[a8]]], Chaps. 10, 11, [[#References|[a6]]], V, Sect. 19.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850146.png" /> denote the free Abelian group whose free basis is given by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850147.png" />-conjugacy classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850148.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850149.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850151.png" /> is a linear character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850152.png" />. Clearly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850153.png" /> is a character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850154.png" /> and hence induction induces an Abelian group homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850155.png" />, which is surjective by 1). Some interesting recent results in [[#References|[a9]]] and [[#References|[a1]]] give explicit (functorial) splittings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850156.png" /> (i.e., an explicit group homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850157.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850158.png" />).
+
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 [[#References|[a9]]] and [[#References|[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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850159.png" /> is the [[Grothendieck group|Grothendieck group]] of the category of finitely generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850160.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850161.png" />, or by considering the modular context, etc., many important analogues of these results emerge, cf. [[#References|[a8]]], Chaps. 12, 16, 17, [[#References|[a5]]], Thm. 2, [[#References|[a10]]].
+
Clearly, $  R ( G ) $
 +
is the [[Grothendieck group|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. [[#References|[a8]]], Chaps. 12, 16, 17, [[#References|[a5]]], Thm. 2, [[#References|[a10]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Boltje,  "A canonical Brauer induction formula"  ''Asterisque'' , '''181/2'''  (1990)  pp. 31–59</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Brauer,  "On Artin's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850162.png" />-series with general group characters"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 502–514</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Brauer,  "A characterization of the characters of a group of finite order"  ''Ann. of Math.'' , '''57'''  (1953)  pp. 357–377</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Brauer,  J. Tate,  "On the characters of finite groups"  ''Ann. of Math.'' , '''62'''  (1955)  pp. 1–7</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Broué,  "Sur l'induction des modules indecomposables et la projectraité relative"  ''Math. Z.'' , '''149'''  (1976)  pp. 227–245</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''I''' , Springer  (1967)  pp. Chapt. V</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I.M. Isaacs,  "Character theory of finite groups" , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.-P. Serre,  "Linear representations of finite groups" , Springer  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  V. Snaith,  "Explicit Brauer induction"  ''Invent. Math.'' , '''94'''  (1988)  pp. 455–478</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  X. Zhou,  "On the decomposition map of Grothendieck groups"  ''Math. Z.'' , '''206'''  (1991)  pp. 533–534</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Boltje,  "A canonical Brauer induction formula"  ''Asterisque'' , '''181/2'''  (1990)  pp. 31–59</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Brauer,  "On Artin's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850162.png" />-series with general group characters"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 502–514</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Brauer,  "A characterization of the characters of a group of finite order"  ''Ann. of Math.'' , '''57'''  (1953)  pp. 357–377</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Brauer,  J. Tate,  "On the characters of finite groups"  ''Ann. of Math.'' , '''62'''  (1955)  pp. 1–7</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Broué,  "Sur l'induction des modules indecomposables et la projectraité relative"  ''Math. Z.'' , '''149'''  (1976)  pp. 227–245</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''I''' , Springer  (1967)  pp. Chapt. V</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I.M. Isaacs,  "Character theory of finite groups" , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.-P. Serre,  "Linear representations of finite groups" , Springer  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  V. Snaith,  "Explicit Brauer induction"  ''Invent. Math.'' , '''94'''  (1988)  pp. 455–478</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  X. Zhou,  "On the decomposition map of Grothendieck groups"  ''Math. Z.'' , '''206'''  (1991)  pp. 533–534</TD></TR></table>

Latest revision as of 06:29, 30 May 2020


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 -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
How to Cite This Entry:
Brauer characterization of characters. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer_characterization_of_characters&oldid=18401
This article was adapted from an original article by M.E. Harris (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article