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''Schur multiplier, of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s0834601.png" />''
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The [[Cohomology group|cohomology group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s0834602.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s0834603.png" /> is the multiplicative group of complex numbers with trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s0834604.png" />-action. The Schur multiplicator was introduced by I. Schur [[#References|[1]]] in his work on finite-dimensional complex projective representations of a group (cf. [[Projective representation|Projective representation]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s0834605.png" /> is such a representation, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s0834606.png" /> can be interpreted as a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s0834607.png" /> such that
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<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/s/s083/s083460/s0834608.png" /></td> </tr></table>
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''Schur multiplier, of a group  $  G $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s0834609.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346010.png" />-cocycle with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346011.png" />. In particular, the projective representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346012.png" /> is the projectivization of a [[Linear representation|linear representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346013.png" /> if and only if the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346014.png" /> determines the trivial element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346017.png" /> is called a closed group in the sense of Schur. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346018.png" /> is a finite group, then there exist natural isomorphisms
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The [[Cohomology group|cohomology group]]  $  H  ^ {2} ( G, \mathbf C  ^  \star  ) $,
 +
where $  \mathbf C  ^  \star  $
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is the multiplicative group of complex numbers with trivial  $  G $-
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action. The Schur multiplicator was introduced by I. Schur [[#References|[1]]] in his work on finite-dimensional complex projective representations of a group (cf. [[Projective representation|Projective representation]]). If $  \rho : G \rightarrow  \mathop{\rm PGL} ( n) $
 +
is such a representation, then $  \rho $
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can be interpreted as a mapping  $  \pi : G \rightarrow  \mathop{\rm GL} ( n) $
 +
such that
  
<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/s/s083/s083460/s08346019.png" /></td> </tr></table>
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$$
 +
\pi ( \sigma ) \pi ( \tau )  = a _ {\sigma , \tau }  \pi ( \sigma , \tau ),
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346020.png" />. If a central extension
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where  $  a _ {\sigma , \tau }  $
 +
is a  $  2 $-
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cocycle with values in  $  \mathbf C  ^  \star  $.
 +
In particular, the projective representation  $  \rho $
 +
is the projectivization of a [[Linear representation|linear representation]]  $  \pi $
 +
if and only if the cocycle  $  a _ {\sigma , \tau }  $
 +
determines the trivial element of the group  $  H  ^ {2} ( G, \mathbf C  ^  \star  ) $.
 +
If  $  H  ^ {2} ( G, \mathbf C  ^  \star  ) = 0 $,
 +
then  $  G $
 +
is called a closed group in the sense of Schur. If $  G $
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is a finite group, then there exist natural isomorphisms
  
<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/s/s083/s083460/s08346021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$
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H  ^ {2} ( G, \mathbf C  ^  \star  )  \cong  H  ^ {2} ( G, \mathbf Q / \mathbf Z )  \cong \
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H  ^ {3} ( G, \mathbf Z ).
 +
$$
  
of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346022.png" /> is given, then there is a natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346023.png" /> whose image coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346024.png" />. This mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346025.png" /> coincides with the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346026.png" /> induced by the cup-product with the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346027.png" /> defined by the extension (*). Conversely, for any subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346028.png" /> there is an extension (*) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346029.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346030.png" />, then the extension (*) is uniquely determined by the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346032.png" /> is a monomorphism, then any projective representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346033.png" /> is induced by some linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083460/s08346034.png" />.
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Let  $  M( G) = H  ^ {-} 3 ( G, \mathbf Z ) =  \mathop{\rm Char} ( H  ^ {3} ( G, \mathbf Z )) $.
 +
If a central extension
 +
 
 +
$$ \tag{* }
 +
1  \rightarrow  A  \rightarrow  F  \rightarrow  G  \rightarrow  1
 +
$$
 +
 
 +
of a finite group $  G $
 +
is given, then there is a natural mapping $  \phi : M( G) \rightarrow A $
 +
whose image coincides with $  A \cap [ F, F  ] $.  
 +
This mapping $  \phi $
 +
coincides with the mapping $  H  ^ {-} 3 ( G, \mathbf Z ) \rightarrow H  ^ {-} 1 ( G, A) $
 +
induced by the cup-product with the element of $  H  ^ {2} ( G, A) $
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defined by the extension (*). Conversely, for any subgroup $  C \subset  M( G) $
 +
there is an extension (*) such that $  \mathop{\rm Ker}  \phi = C $.  
 +
If $  G = [ G, G] $,  
 +
then the extension (*) is uniquely determined by the homomorphism $  \phi $.  
 +
If $  \phi $
 +
is a monomorphism, then any projective representation of $  G $
 +
is induced by some linear representation of $  F $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Schur,  "Ueber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen"  ''J. Reine Angew. Math.'' , '''127'''  (1904)  pp. 20–50</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Schur,  "Ueber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen"  ''J. Reine Angew. Math.'' , '''127'''  (1904)  pp. 20–50</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1975)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Gruenberg,  "Cohomological topics in group theory" , ''Lect. notes in math.'' , '''143''' , Springer  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Methods of representation theory" , '''I''' , Wiley (Interscience)  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Gruenberg,  "Cohomological topics in group theory" , ''Lect. notes in math.'' , '''143''' , Springer  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Methods of representation theory" , '''I''' , Wiley (Interscience)  (1981)</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


Schur multiplier, of a group $ G $

The cohomology group $ H ^ {2} ( G, \mathbf C ^ \star ) $, where $ \mathbf C ^ \star $ is the multiplicative group of complex numbers with trivial $ G $- action. The Schur multiplicator was introduced by I. Schur [1] in his work on finite-dimensional complex projective representations of a group (cf. Projective representation). If $ \rho : G \rightarrow \mathop{\rm PGL} ( n) $ is such a representation, then $ \rho $ can be interpreted as a mapping $ \pi : G \rightarrow \mathop{\rm GL} ( n) $ such that

$$ \pi ( \sigma ) \pi ( \tau ) = a _ {\sigma , \tau } \pi ( \sigma , \tau ), $$

where $ a _ {\sigma , \tau } $ is a $ 2 $- cocycle with values in $ \mathbf C ^ \star $. In particular, the projective representation $ \rho $ is the projectivization of a linear representation $ \pi $ if and only if the cocycle $ a _ {\sigma , \tau } $ determines the trivial element of the group $ H ^ {2} ( G, \mathbf C ^ \star ) $. If $ H ^ {2} ( G, \mathbf C ^ \star ) = 0 $, then $ G $ is called a closed group in the sense of Schur. If $ G $ is a finite group, then there exist natural isomorphisms

$$ H ^ {2} ( G, \mathbf C ^ \star ) \cong H ^ {2} ( G, \mathbf Q / \mathbf Z ) \cong \ H ^ {3} ( G, \mathbf Z ). $$

Let $ M( G) = H ^ {-} 3 ( G, \mathbf Z ) = \mathop{\rm Char} ( H ^ {3} ( G, \mathbf Z )) $. If a central extension

$$ \tag{* } 1 \rightarrow A \rightarrow F \rightarrow G \rightarrow 1 $$

of a finite group $ G $ is given, then there is a natural mapping $ \phi : M( G) \rightarrow A $ whose image coincides with $ A \cap [ F, F ] $. This mapping $ \phi $ coincides with the mapping $ H ^ {-} 3 ( G, \mathbf Z ) \rightarrow H ^ {-} 1 ( G, A) $ induced by the cup-product with the element of $ H ^ {2} ( G, A) $ defined by the extension (*). Conversely, for any subgroup $ C \subset M( G) $ there is an extension (*) such that $ \mathop{\rm Ker} \phi = C $. If $ G = [ G, G] $, then the extension (*) is uniquely determined by the homomorphism $ \phi $. If $ \phi $ is a monomorphism, then any projective representation of $ G $ is induced by some linear representation of $ F $.

References

[1] I. Schur, "Ueber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen" J. Reine Angew. Math. , 127 (1904) pp. 20–50
[2] S. MacLane, "Homology" , Springer (1975)

Comments

References

[a1] G. Gruenberg, "Cohomological topics in group theory" , Lect. notes in math. , 143 , Springer (1970)
[a2] C.W. Curtis, I. Reiner, "Methods of representation theory" , I , Wiley (Interscience) (1981)
How to Cite This Entry:
Schur multiplicator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_multiplicator&oldid=48625
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article