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Concept
Projective representation
Summary
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) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation).
In more concrete terms, a projective representation of is a collection of operators satisfying the homomorphism property up to a constant:
for some constant . Equivalently, a projective representation of is a collection of operators , such that . Note that, in this notation, is a set of linear operators related by multiplication with some nonzero scalar.
If it is possible to choose a particular representative in each family of operators in such a way that the homomorphism property is satisfied on the nose, rather than just up to a constant, then we say that can be "de-projectivized", or that can be "lifted to an ordinary representation". More concretely, we thus say that can be de-projectivized if there are for each such that . This possibility is discussed further below.
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 an ordinary linear representation. A general projective representation ρ: G → PGL(V) cannot be lifted to a linear representation G → GL(V), and the obstruction to this lifting can be understood via group cohomology, as described below.
However, one can lift a projective representation of G to a linear representation of a different group H, which will be a central extension of G. The group is the subgroup of defined as follows:
where is the quotient map of onto .
Official source
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