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''of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p0753201.png" />''
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A homomorphism of this group into the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p0753202.png" /> of projective transformations of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p0753203.png" /> associated to a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p0753204.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p0753205.png" />.
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{{TEX|done}}
  
With each projective representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p0753206.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p0753207.png" /> there is associated a central extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p0753208.png" />: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p0753209.png" /> be the [[General linear group|general linear group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532010.png" />. Then one has a natural [[Exact sequence|exact sequence]]
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''of a group $  G $''
  
<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/p/p075/p075320/p07532011.png" /></td> </tr></table>
+
A homomorphism of this group into the group  $  \mathop{\rm PGL} ( V ) $
 +
of projective transformations of the projective space  $  P ( V ) $
 +
associated to a vector space  $  V $
 +
over a field  $  k $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532012.png" /> is the natural projection of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532013.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532015.png" /> is the imbedding of the multiplicative group of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532016.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532017.png" /> by scalar matrices. The pullback along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532018.png" /> gives rise to the following commutative diagram with exact rows:
+
With each projective representation  $  \phi $
 +
of the group $  G $
 +
there is associated a central extension of $  G $:
 +
Let  $  \mathop{\rm GL} ( V) $
 +
be the [[General linear group|general linear group]] of $  V $.  
 +
Then one has a natural [[Exact sequence|exact sequence]]
  
<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/p/p075/p075320/p07532019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$
 +
1  \rightarrow  k  ^  \times  \stackrel{i}\rightarrow  \mathop{\rm GL} ( V)  \stackrel{p}\rightarrow  \
 +
\mathop{\rm PGL} ( V) \rightarrow  1,
 +
$$
  
which is the associated central extension. Every section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532020.png" />, i.e. homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532022.png" />, has the property
+
where  $  p $
 +
is the natural projection of the group  $  \mathop{\rm GL} ( V) $
 +
onto  $  \mathop{\rm PGL} ( V) $
 +
and  $  i $
 +
is the imbedding of the multiplicative group of the field  $  k $
 +
into  $  \mathop{\rm GL} ( V) $
 +
by scalar matrices. The pullback along  $  \phi $
 +
gives rise to the following commutative diagram with exact rows:
  
<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/p/p075/p075320/p07532023.png" /></td> </tr></table>
+
$$ \tag{* }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532024.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532025.png" />-cocycle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532026.png" />. The cohomology class of this cocycle is independent of the choice of the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532027.png" />. It is determined by the projective representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532028.png" /> and determines the equivalence class of the extension (*). The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532029.png" /> is necessary and sufficient for the projective representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532030.png" /> to be the composition of a linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532031.png" /> with the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532032.png" />.
+
\begin{array}{ccccccccc}
 +
1  &{ \rightarrow  }  &{k  ^  \times  }  &{  \mathop \rightarrow \limits ^ { {i }}  }  & \mathop{\rm GL} ( V)  &{  \mathop \rightarrow \limits ^ { {p }}  }  & \mathop{\rm PGL} ( V) &{ \rightarrow  }  & 1  \\
 +
{}  &{}  &\|  &{}  &{\uparrow { {\widehat \phi  } } }  &{}  &{\uparrow  \phi }  &{}  &{}  \\
 +
1  &{ \rightarrow  }  &{k  ^  \times  }  &{  \mathop \rightarrow \limits _ { {\widehat{i}  }}  }  &{E _  \phi  }  &{  \mathop \rightarrow \limits _ { {{\widehat{p}  }}  } }  & G  &{ \rightarrow  }  & 1  \\
 +
\end{array}
  
Projective representations arise naturally in studying linear representations of group extensions. The most important examples of projective representations are: the spinor representation of an orthogonal group and the Weyl representation of a symplectic group. The definitions of equivalence and irreducibility of representations carry over directly to projective representations. The classification of the irreducible projective representations of finite groups was obtained by I. Schur (1904).
+
$$
 
 
A projective representation is said to be unitary if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532033.png" /> is a Hilbert space and if the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532034.png" /> can be chosen so that it takes values in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532035.png" /> of unitary operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532036.png" />. Irreducible unitary projective representations of topological groups have been studied [[#References|[4]]]; for a connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532037.png" /> this study reduces to a study of the irreducible unitary representations of a simply-connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532038.png" />, the Lie algebra of which is the central extension of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532039.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532040.png" /> by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532041.png" />-dimensional commutative Lie algebra, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075320/p07532042.png" />.
 
  
====References====
+
which is the associated central extension. Every section  $ \psi : G \rightarrow E _ \phi $,
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Kirillov,  "Elements of the theory of representations" , Springer (1976) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.W. Mackey,  "Unitary representations of group extensions, I" ''Acta Math.'' , '''99''' (1958) pp. 265–311</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V. Bargmann,   "Irreducible unitary representations of the Lorentz group"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 568–640</TD></TR></table>
+
i.e. homomorphism $ \psi $
 +
such that $ \psi \cdot \widehat{p} \mathop{\rm id} _ {G} $,  
 +
has the property
  
 +
$$
 +
\psi ( g _ {1} g _ {2} )  =  c ( g _ {1} , g _ {2} )
 +
\psi ( g _ {1} ) \psi ( g _ {2} ) ,
 +
$$
  
 +
where  $  c:  G \times G \rightarrow k  ^  \times  $
 +
is a  $  2 $-
 +
cocycle of  $  G $.
 +
The cohomology class of this cocycle is independent of the choice of the section  $  s $.
 +
It is determined by the projective representation  $  \phi $
 +
and determines the equivalence class of the extension (*). The condition  $  h = 0 $
 +
is necessary and sufficient for the projective representation  $  \phi $
 +
to be the composition of a linear representation of  $  G $
 +
with the projection  $  p $.
  
====Comments====
+
Projective representations arise naturally in studying linear representations of group extensions. The most important examples of projective representations are: the spinor representation of an orthogonal group and the Weyl representation of a symplectic group. The definitions of equivalence and irreducibility of representations carry over directly to projective representations. The classification of the irreducible projective representations of finite groups was obtained by I. Schur (1904).
  
 +
A projective representation is said to be unitary if  $  V $
 +
is a Hilbert space and if the mapping  $  \psi $
 +
can be chosen so that it takes values in the group  $  U ( V ) $
 +
of unitary operators on  $  V $.
 +
Irreducible unitary projective representations of topological groups have been studied [[#References|[4]]]; for a connected Lie group  $  G $
 +
this study reduces to a study of the irreducible unitary representations of a simply-connected Lie group  $  \widetilde{G}  $,
 +
the Lie algebra of which is the central extension of the Lie algebra  $  \mathfrak g $
 +
of the group  $  G $
 +
by a  $  d $-
 +
dimensional commutative Lie algebra, where  $  d =  \mathop{\rm dim}  H  ^ {2} ( \mathfrak g , \mathbf R ) $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,  "Methods of representation theory" , '''1–2''' , Wiley (Interscience)  (1981–1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.M. Isaacs,   "Character theory of finite groups" , Acad. Press  (1976)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov, "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis, I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  G.W. Mackey, "Unitary representations of group extensions, I"  ''Acta Math.'' , '''99'''  (1958)  pp. 265–311</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  V. Bargmann, "Irreducible unitary representations of the Lorentz group"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 568–640</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C.W. Curtis, I. Reiner,  "Methods of representation theory" , '''1–2''' , Wiley (Interscience)  (1981–1987)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  I.M. Isaacs, "Character theory of finite groups" , Acad. Press  (1976)</TD></TR></table>

Latest revision as of 18:02, 13 April 2023


of a group $ G $

A homomorphism of this group into the group $ \mathop{\rm PGL} ( V ) $ of projective transformations of the projective space $ P ( V ) $ associated to a vector space $ V $ over a field $ k $.

With each projective representation $ \phi $ of the group $ G $ there is associated a central extension of $ G $: Let $ \mathop{\rm GL} ( V) $ be the general linear group of $ V $. Then one has a natural exact sequence

$$ 1 \rightarrow k ^ \times \stackrel{i}\rightarrow \mathop{\rm GL} ( V) \stackrel{p}\rightarrow \ \mathop{\rm PGL} ( V) \rightarrow 1, $$

where $ p $ is the natural projection of the group $ \mathop{\rm GL} ( V) $ onto $ \mathop{\rm PGL} ( V) $ and $ i $ is the imbedding of the multiplicative group of the field $ k $ into $ \mathop{\rm GL} ( V) $ by scalar matrices. The pullback along $ \phi $ gives rise to the following commutative diagram with exact rows:

$$ \tag{* } \begin{array}{ccccccccc} 1 &{ \rightarrow } &{k ^ \times } &{ \mathop \rightarrow \limits ^ { {i }} } & \mathop{\rm GL} ( V) &{ \mathop \rightarrow \limits ^ { {p }} } & \mathop{\rm PGL} ( V) &{ \rightarrow } & 1 \\ {} &{} &\| &{} &{\uparrow { {\widehat \phi } } } &{} &{\uparrow \phi } &{} &{} \\ 1 &{ \rightarrow } &{k ^ \times } &{ \mathop \rightarrow \limits _ { {\widehat{i} }} } &{E _ \phi } &{ \mathop \rightarrow \limits _ { {{\widehat{p} }} } } & G &{ \rightarrow } & 1 \\ \end{array} $$

which is the associated central extension. Every section $ \psi : G \rightarrow E _ \phi $, i.e. homomorphism $ \psi $ such that $ \psi \cdot \widehat{p} = \mathop{\rm id} _ {G} $, has the property

$$ \psi ( g _ {1} g _ {2} ) = c ( g _ {1} , g _ {2} ) \psi ( g _ {1} ) \psi ( g _ {2} ) , $$

where $ c: G \times G \rightarrow k ^ \times $ is a $ 2 $- cocycle of $ G $. The cohomology class of this cocycle is independent of the choice of the section $ s $. It is determined by the projective representation $ \phi $ and determines the equivalence class of the extension (*). The condition $ h = 0 $ is necessary and sufficient for the projective representation $ \phi $ to be the composition of a linear representation of $ G $ with the projection $ p $.

Projective representations arise naturally in studying linear representations of group extensions. The most important examples of projective representations are: the spinor representation of an orthogonal group and the Weyl representation of a symplectic group. The definitions of equivalence and irreducibility of representations carry over directly to projective representations. The classification of the irreducible projective representations of finite groups was obtained by I. Schur (1904).

A projective representation is said to be unitary if $ V $ is a Hilbert space and if the mapping $ \psi $ can be chosen so that it takes values in the group $ U ( V ) $ of unitary operators on $ V $. Irreducible unitary projective representations of topological groups have been studied [4]; for a connected Lie group $ G $ this study reduces to a study of the irreducible unitary representations of a simply-connected Lie group $ \widetilde{G} $, the Lie algebra of which is the central extension of the Lie algebra $ \mathfrak g $ of the group $ G $ by a $ d $- dimensional commutative Lie algebra, where $ d = \mathop{\rm dim} H ^ {2} ( \mathfrak g , \mathbf R ) $.

References

[1] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[2] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[3] G.W. Mackey, "Unitary representations of group extensions, I" Acta Math. , 99 (1958) pp. 265–311
[4] V. Bargmann, "Irreducible unitary representations of the Lorentz group" Ann. of Math. , 48 (1947) pp. 568–640
[a1] C.W. Curtis, I. Reiner, "Methods of representation theory" , 1–2 , Wiley (Interscience) (1981–1987)
[a2] I.M. Isaacs, "Character theory of finite groups" , Acad. Press (1976)
How to Cite This Entry:
Projective representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_representation&oldid=12442
This article was adapted from an original article by A.A. Kirillov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article