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.
Linear representations and projective representations
One way in which a projective representation can arise is by taking a linear group representation of on and applying the quotient map
which is the quotient by the subgroup 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 cannot be lifted to a linear representation , and the obstruction to this lifting can be understood via group cohomology, as described below.
However, one can lift a projective representation of to a linear representation of a different group , which will be a central extension of . The group is the subgroup of defined as follows: , where is the quotient map of onto . Since is a homomorphism, it is easy to check that is, indeed, a subgroup of . If the original projective representation is faithful, then is isomorphic to the preimage in of .
We can define a homomorphism by setting . The kernel of is: , which is contained in the center of . It is clear also that is surjective, so that is a central extension of . We can also define an ordinary representation of by setting . The ordinary representation of is a lift of the projective representation of in the sense that: .
If is a perfect group there is a single universal perfect central extension of that can be used.
Group cohomology
The analysis of the lifting question involves group cohomology. Indeed, if one fixes for each in a lifted element in lifting from back to , the lifts then satisfy
for some scalar in . It follows that the 2-cocycle or Schur multiplier satisfies the cocycle equation
for all in . This depends on the choice of the lift ; a different choice of lift will result in a different cocycle
cohomologous to . Thus defines a unique class in . This class might not be trivial. For example, in the case of the symmetric group and alternating group, Schur established that there is exactly one non-trivial class of Schur multiplier, and completely determined all the corresponding irreducible representations.
In general, a nontrivial class leads to an extension problem for . If is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down to . The solution is always a central extension. From Schur's lemma, it follows that the irreducible representations of central extensions of , and the irreducible projective representations of , are essentially the same objects.
First example: discrete Fourier transform
Consider the field of integers mod , where is prime, and let be the -dimensional space of functions on with values in . For each in , define two operators, and on as follows:
We write the formula for as if and were integers, but it is easily seen that the result only depends on the value of and mod . The operator is a translation, while is a shift in frequency space (that is, it has the effect of translating the discrete Fourier transform of ).
One may easily verify that for any and in , the operators and commute up to multiplication by a constant: .