is a set of linear operators related by multiplication with some nonzero scalar.
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
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).
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.
The analysis of the lifting question involves group cohomology.
It follows that the 2-cocycle or Schur multiplier c satisfies the cocycle equation for all g, h, k in G. This c depends on the choice of the lift L; a different choice of lift L′(g) = f(g) L(g) will result in a different cocycle cohomologous to c. Thus L defines a unique class in H2(G, F∗).
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.
[2] In general, a nontrivial class leads to an extension problem for G. If G is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down to G. The solution is always a central extension.
is a shift in frequency space (that is, it has the effect of translating the discrete Fourier transform of
commute up to multiplication by a constant: We may therefore define a projective representation
In many cases of interest it suffices to consider representations of covering groups.
The preceding construction is an important source of examples of projective representations.
Bargmann's theorem (discussed below) gives a criterion under which every irreducible projective unitary representation of
A physically important example of the above construction comes from the case of the rotation group SO(3), whose universal cover is SU(2).
Notable cases of covering groups giving interesting projective representations: In quantum physics, symmetry of a physical system is typically implemented by means of a projective unitary representation
on the quantum Hilbert space, that is, a continuous homomorphism where
[That is to say, the space of (pure) states is the set of equivalence classes of unit vectors, where two unit vectors are considered equivalent if they are proportional.]
In the finite-dimensional case, it is always possible to "de-projectivize" the Lie-algebra representation
[4] In light of the homomorphisms theorem, it is then possible to de-projectivize
Thus, there is a natural one-to-one correspondence between the irreducible projective representations of
An important example is the case of SO(3), whose universal cover is SU(2).
The results of the previous subsection do not hold in the infinite-dimensional case, simply because the trace of
But no choice of the phase factors will lead to an ordinary unitary representation, since translations in position do not commute with translations in momentum (and multiplying by a nonzero constant will not change this).
These operators do, however, come from an ordinary unitary representation of the Heisenberg group, which is a one-dimensional central extension of
On the other hand, Bargmann's theorem states that if the second Lie algebra cohomology group
[9][10] More precisely, suppose we begin with a projective unitary representation
—as the previous example shows—because the second cohomology group of the associated commutative Lie algebra is nontrivial.
This last result is important for Wigner's classification of the projective unitary representations of the Poincaré group.
The proof of Bargmann's theorem goes by considering a central extension
may be identified with the space of one-dimensional (again, in the aforementioned sense) central extensions of