# 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)