Projective representation

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)