Projective representation

In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group

where GL(V, F) is the general linear group of invertible linear transformations of V over F and F^∗ is the normal subgroup consisting of multiplications of vectors in V by nonzero elements of F (that is, scalar multiples of the identity; scalar transformations).

One way in which a projective representation can arise is by taking a linear group representation of $G$ on $V$ and applying the quotient map

which is the quotient by the subgroup $F *$ of scalar transformations (diagonal matrices with all diagonal entries equal). The interest for algebra is in the process in the other direction: given a projective representation, try to 'lift' it to a conventional linear representation.

In general, given a projective representation $ρ : G \to PGL(V)$ it cannot be lifted to a linear representation $G \to GL(V)$ , and the obstruction to this lifting can be understood via group homology, as described below. However, one can lift a projective representation of $G$ to a linear representation of a different group $C$ , which will be a central extension of $G$ . To understand this, note that $GL(V) \to PGL(V)$ is a central extension of $PGL$ , meaning that the kernel is central (in fact, is exactly the center of $GL$ ). One can pull back the projective representation $ρ : G \to PGL(V)$ along the quotient map, obtaining a linear representation $σ : C \to GL(V)$ and $C$ will be a central extension of $G$ because it is a pullback of a central extension. Thus projective representations of $G$ can be understood in terms of linear representations of (certain) central extensions of $G$ . Notably, for $G$ a perfect group there is a single universal perfect central extension of $G$ that can be used.

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