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''(for group representations)''
 
''(for group representations)''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c1102601.png" /> be a [[Normal subgroup|normal subgroup]] of a [[Finite group|finite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c1102602.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c1102603.png" /> be the [[Group algebra|group algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c1102604.png" /> over a [[Commutative ring|commutative ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c1102605.png" />. Given an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c1102606.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c1102607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c1102608.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c1102609.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026010.png" />-module whose underlying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026011.png" />-module is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026012.png" /> and on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026013.png" /> acts according to the rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026016.png" /> denotes the module operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026018.png" /> the operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026019.png" />. By definition, the inertia group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026021.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026022.png" />. It is clear that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026023.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026024.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026025.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026026.png" />, it is customary to say that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026027.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026029.png" />-invariant
+
Let $  N $
 +
be a [[Normal subgroup|normal subgroup]] of a [[Finite group|finite group]] $  G $
 +
and let $  RG $
 +
be the [[Group algebra|group algebra]] of $  G $
 +
over a [[Commutative ring|commutative ring]] $  R $.  
 +
Given an $  RN $-
 +
module $  V $
 +
and $  g \in G $,  
 +
let $  ^ {g} V $
 +
be the $  RN $-
 +
module whose underlying $  R $-
 +
module is $  V $
 +
and on which $  N $
 +
acts according to the rule $  n * v = ( g ^ {- 1 } ng ) v $,  
 +
$  v \in V $,  
 +
where $  n * v $
 +
denotes the module operation in $  ^ {g} V $
 +
and $  nv $
 +
the operation in $  V $.  
 +
By definition, the inertia group $  H $
 +
of $  V $
 +
is $  H = \{ {g \in G } : {V \cong  ^ {g} V } \} $.  
 +
It is clear that $  H $
 +
is a subgroup of $  G $
 +
containing $  N $;  
 +
if $  H = G $,  
 +
it is customary to say that $  V $
 +
is $  G $-
 +
invariant
  
Important information concerning simple and indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026030.png" />-modules can be obtained by applying (perhaps repeatedly) three basic operations:
+
Important information concerning simple and indecomposable $  RG $-
 +
modules can be obtained by applying (perhaps repeatedly) three basic operations:
  
i) restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026031.png" />;
+
i) restriction to $  RN $;
  
ii) extension from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026032.png" />; and
+
ii) extension from $  RN $;  
 +
and
  
iii) induction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026033.png" />. This is the content of the so-called Clifford theory, which was originally developed by A.H. Clifford (see [[#References|[a1]]]) for the classical case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026034.png" /> is a [[Field|field]]. General references for this area are [[#References|[a2]]], [[#References|[a3]]].
+
iii) induction from $  RN $.  
 +
This is the content of the so-called Clifford theory, which was originally developed by A.H. Clifford (see [[#References|[a1]]]) for the classical case where $  R $
 +
is a [[Field|field]]. General references for this area are [[#References|[a2]]], [[#References|[a3]]].
  
 
The most important results are as follows.
 
The most important results are as follows.
  
 
==Restriction to normal subgroups of representations.==
 
==Restriction to normal subgroups of representations.==
Given a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026036.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026037.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026038.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026039.png" /> denote the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026040.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026041.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026042.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026043.png" />-module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026044.png" /> denotes the induced module. For any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026045.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026046.png" /> be the direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026047.png" /> copies of a given module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026048.png" />. A classical Clifford theorem, originally proved for the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026049.png" /> is a field, holds for an arbitrary commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026050.png" /> and asserts the following. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026051.png" /> is a simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026052.png" />-module. Then there exists a simple submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026054.png" />; for any such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026055.png" /> and the inertia group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026057.png" />, the following properties hold.
+
Given a subgroup $  H $
 +
of $  G $
 +
and an $  RG $-
 +
module $  U $,  
 +
let $  U _ {H} $
 +
denote the restriction of $  U $
 +
to $  RH $.  
 +
If $  V $
 +
is an $  RH $-
 +
module, then $  V  ^ {G} $
 +
denotes the induced module. For any integer $  e \geq  1 $,  
 +
let $  eV $
 +
be the direct sum of $  e $
 +
copies of a given module $  V $.  
 +
A classical Clifford theorem, originally proved for the case where $  R $
 +
is a field, holds for an arbitrary commutative ring $  R $
 +
and asserts the following. Assume that $  U $
 +
is a simple $  RG $-
 +
module. Then there exists a simple submodule $  V $
 +
of $  U _ {N} $;  
 +
for any such $  V $
 +
and the inertia group $  H $
 +
of $  V $,  
 +
the following properties hold.
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026059.png" /> is a left transversal for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026061.png" />. Moreover, the modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026063.png" />, are pairwise non-isomorphic simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026064.png" />-modules.
+
a) $  U _ {N} \cong e ( \oplus _ {t \in T }  ^ {t} V ) $,  
 +
where $  T $
 +
is a left transversal for $  H $
 +
in $  G $.  
 +
Moreover, the modules $  ^ {t} V $,  
 +
$  t \in T $,  
 +
are pairwise non-isomorphic simple $  RN $-
 +
modules.
  
b) The sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026065.png" /> of all submodules of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026066.png" /> isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026067.png" /> is a simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026068.png" />-module such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026070.png" />.
+
b) The sum $  W $
 +
of all submodules of $  U _ {N} $
 +
isomorphic to $  V $
 +
is a simple $  RH $-
 +
module such that $  W _ {N} \cong eV $
 +
and $  U \cong W  ^ {G} $.
  
The above result holds in the more general case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026071.png" /> is a finite group. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026072.png" /> is infinite, then Clifford's theorem is no longer true (see [[#References|[a3]]]).
+
The above result holds in the more general case where $  {G / N } $
 +
is a finite group. However, if $  {G / N } $
 +
is infinite, then Clifford's theorem is no longer true (see [[#References|[a3]]]).
  
 
==Induction from normal subgroups of representations.==
 
==Induction from normal subgroups of representations.==
The principal result concerning induction is the Green indecomposable theorem, described below. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026073.png" /> is a complete [[Local ring|local ring]] and a principal ideal domain (cf. also [[Principal ideal ring|Principal ideal ring]]). An integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026074.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026075.png" /> is called an extension, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026076.png" />, written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026077.png" />, if the following conditions hold:
+
The principal result concerning induction is the Green indecomposable theorem, described below. Assume that $  R $
 +
is a complete [[Local ring|local ring]] and a principal ideal domain (cf. also [[Principal ideal ring|Principal ideal ring]]). An integral domain $  S $
 +
containing $  R $
 +
is called an extension, of $  R $,  
 +
written $  {S / R } $,  
 +
if the following conditions hold:
  
A) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026078.png" /> is a principal ideal domain and a local ring;
+
A) $  S $
 +
is a principal ideal domain and a local ring;
  
B) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026079.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026080.png" />-free;
+
B) $  S $
 +
is $  R $-
 +
free;
  
C) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026081.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026082.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026083.png" /> is finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026084.png" /> is a finitely generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026085.png" />-module. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026086.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026087.png" /> is said to be absolutely indecomposable if for every finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026089.png" /> is an indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026090.png" />-module.
+
C) $  J ( S )  ^ {e} = J ( R ) S $
 +
for some integer $  e \geq  1 $.  
 +
One says that $  {S / R } $
 +
is finite if $  S $
 +
is a finitely generated $  R $-
 +
module. An $  RG $-
 +
module $  V $
 +
is said to be absolutely indecomposable if for every finite extension $  {S / R } $,  
 +
$  S \otimes _ {R} V $
 +
is an indecomposable $  SG $-
 +
module.
  
Assume that the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026091.png" /> is of prime characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026092.png" /> (cf. also [[Characteristic of a field|Characteristic of a field]]) and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026093.png" /> is a [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026094.png" />-group]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026095.png" /> is a finitely generated absolutely indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026096.png" />-module, then the induced module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026097.png" /> is absolutely indecomposable. Green's original statement pertained to the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026098.png" /> is a field. A proof in full generality is contained in [[#References|[a3]]].
+
Assume that the field $  {R / {J ( R ) } } $
 +
is of prime characteristic $  p $(
 +
cf. also [[Characteristic of a field|Characteristic of a field]]) and that $  {G / N } $
 +
is a [[P-group| $  p $-
 +
group]]. If $  V $
 +
is a finitely generated absolutely indecomposable $  RN $-
 +
module, then the induced module $  V  ^ {G} $
 +
is absolutely indecomposable. Green's original statement pertained to the case where $  R $
 +
is a field. A proof in full generality is contained in [[#References|[a3]]].
  
 
==Extension from normal subgroups of representations.==
 
==Extension from normal subgroups of representations.==
The best result to date (1996) is Isaacs theorem, described below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c11026099.png" /> be a normal [[Hall subgroup|Hall subgroup]] of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260100.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260101.png" /> be an arbitrary commutative ring and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260102.png" /> be a simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260103.png" />-invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260104.png" />-module. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260105.png" /> extends to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260106.png" />-module, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260107.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260108.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260109.png" />. Originally, R. Isaacs proved only the special case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110260/c110260110.png" /> is a field. A proof in full generality can be found in [[#References|[a3]]].
+
The best result to date (1996) is Isaacs theorem, described below. Let $  N $
 +
be a normal [[Hall subgroup|Hall subgroup]] of a finite group $  G $,  
 +
let $  R $
 +
be an arbitrary commutative ring and let $  V $
 +
be a simple $  G $-
 +
invariant $  RN $-
 +
module. Then $  V $
 +
extends to an $  RG $-
 +
module, i.e. $  V \cong U _ {N} $
 +
for some $  RG $-
 +
module $  U $.  
 +
Originally, R. Isaacs proved only the special case where $  R $
 +
is a field. A proof in full generality can be found in [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.H. Clifford,  "Representations induced in an invariant subgroup"  ''Ann. of Math. (2)'' , '''38'''  pp. 533–550</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Karpilovsky,  "Clifford theory for group representations" , North-Holland  (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''3''' , North-Holland  (1994)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.H. Clifford,  "Representations induced in an invariant subgroup"  ''Ann. of Math. (2)'' , '''38'''  pp. 533–550</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Karpilovsky,  "Clifford theory for group representations" , North-Holland  (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''3''' , North-Holland  (1994)</TD></TR></table>

Latest revision as of 17:44, 4 June 2020


(for group representations)

Let $ N $ be a normal subgroup of a finite group $ G $ and let $ RG $ be the group algebra of $ G $ over a commutative ring $ R $. Given an $ RN $- module $ V $ and $ g \in G $, let $ ^ {g} V $ be the $ RN $- module whose underlying $ R $- module is $ V $ and on which $ N $ acts according to the rule $ n * v = ( g ^ {- 1 } ng ) v $, $ v \in V $, where $ n * v $ denotes the module operation in $ ^ {g} V $ and $ nv $ the operation in $ V $. By definition, the inertia group $ H $ of $ V $ is $ H = \{ {g \in G } : {V \cong ^ {g} V } \} $. It is clear that $ H $ is a subgroup of $ G $ containing $ N $; if $ H = G $, it is customary to say that $ V $ is $ G $- invariant

Important information concerning simple and indecomposable $ RG $- modules can be obtained by applying (perhaps repeatedly) three basic operations:

i) restriction to $ RN $;

ii) extension from $ RN $; and

iii) induction from $ RN $. This is the content of the so-called Clifford theory, which was originally developed by A.H. Clifford (see [a1]) for the classical case where $ R $ is a field. General references for this area are [a2], [a3].

The most important results are as follows.

Restriction to normal subgroups of representations.

Given a subgroup $ H $ of $ G $ and an $ RG $- module $ U $, let $ U _ {H} $ denote the restriction of $ U $ to $ RH $. If $ V $ is an $ RH $- module, then $ V ^ {G} $ denotes the induced module. For any integer $ e \geq 1 $, let $ eV $ be the direct sum of $ e $ copies of a given module $ V $. A classical Clifford theorem, originally proved for the case where $ R $ is a field, holds for an arbitrary commutative ring $ R $ and asserts the following. Assume that $ U $ is a simple $ RG $- module. Then there exists a simple submodule $ V $ of $ U _ {N} $; for any such $ V $ and the inertia group $ H $ of $ V $, the following properties hold.

a) $ U _ {N} \cong e ( \oplus _ {t \in T } ^ {t} V ) $, where $ T $ is a left transversal for $ H $ in $ G $. Moreover, the modules $ ^ {t} V $, $ t \in T $, are pairwise non-isomorphic simple $ RN $- modules.

b) The sum $ W $ of all submodules of $ U _ {N} $ isomorphic to $ V $ is a simple $ RH $- module such that $ W _ {N} \cong eV $ and $ U \cong W ^ {G} $.

The above result holds in the more general case where $ {G / N } $ is a finite group. However, if $ {G / N } $ is infinite, then Clifford's theorem is no longer true (see [a3]).

Induction from normal subgroups of representations.

The principal result concerning induction is the Green indecomposable theorem, described below. Assume that $ R $ is a complete local ring and a principal ideal domain (cf. also Principal ideal ring). An integral domain $ S $ containing $ R $ is called an extension, of $ R $, written $ {S / R } $, if the following conditions hold:

A) $ S $ is a principal ideal domain and a local ring;

B) $ S $ is $ R $- free;

C) $ J ( S ) ^ {e} = J ( R ) S $ for some integer $ e \geq 1 $. One says that $ {S / R } $ is finite if $ S $ is a finitely generated $ R $- module. An $ RG $- module $ V $ is said to be absolutely indecomposable if for every finite extension $ {S / R } $, $ S \otimes _ {R} V $ is an indecomposable $ SG $- module.

Assume that the field $ {R / {J ( R ) } } $ is of prime characteristic $ p $( cf. also Characteristic of a field) and that $ {G / N } $ is a $ p $- group. If $ V $ is a finitely generated absolutely indecomposable $ RN $- module, then the induced module $ V ^ {G} $ is absolutely indecomposable. Green's original statement pertained to the case where $ R $ is a field. A proof in full generality is contained in [a3].

Extension from normal subgroups of representations.

The best result to date (1996) is Isaacs theorem, described below. Let $ N $ be a normal Hall subgroup of a finite group $ G $, let $ R $ be an arbitrary commutative ring and let $ V $ be a simple $ G $- invariant $ RN $- module. Then $ V $ extends to an $ RG $- module, i.e. $ V \cong U _ {N} $ for some $ RG $- module $ U $. Originally, R. Isaacs proved only the special case where $ R $ is a field. A proof in full generality can be found in [a3].

References

[a1] A.H. Clifford, "Representations induced in an invariant subgroup" Ann. of Math. (2) , 38 pp. 533–550
[a2] G. Karpilovsky, "Clifford theory for group representations" , North-Holland (1989)
[a3] G. Karpilovsky, "Group representations" , 3 , North-Holland (1994)
How to Cite This Entry:
Clifford theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clifford_theory&oldid=46360
This article was adapted from an original article by G. Karpilovsky (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article