Projective representation

of a group

A homomorphism of this group into the group of projective transformations of the projective space associated to a vector space over a field .

With each projective representation of the group there is associated a central extension of : Let be the general linear group of . Then one has a natural exact sequence

where is the natural projection of the group onto and is the imbedding of the multiplicative group of the field into by scalar matrices. The pullback along gives rise to the following commutative diagram with exact rows:

(*)

which is the associated central extension. Every section , i.e. homomorphism such that , has the property

where is a -cocycle of . The cohomology class of this cocycle is independent of the choice of the section . It is determined by the projective representation and determines the equivalence class of the extension (*). The condition is necessary and sufficient for the projective representation to be the composition of a linear representation of with the projection .

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 is a Hilbert space and if the mapping can be chosen so that it takes values in the group of unitary operators on . Irreducible unitary projective representations of topological groups have been studied [4]; for a connected Lie group this study reduces to a study of the irreducible unitary representations of a simply-connected Lie group , the Lie algebra of which is the central extension of the Lie algebra of the group by a -dimensional commutative Lie algebra, where .

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

References

[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)