Schur multiplicator

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)

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)