Projective representation
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In mathematics, in particular in group theory, if G is a group and V is a vector space over a field F, then a projective representation is a group homomorphism from G to
- Aut(V)/F*
where F* here is the normal subgroup of Aut(V) consisting of multiplications of vectors in V by nonzero elements of F (that is, scalar multiples of the identity), and Aut(V) means the automorphism group of the vector space underlying V. This may be briefly described otherwise, as a homomorphism to a projective linear group
- PGL(V).
One way in which such representations can arise is using the homomorphism
- GL(V) → PGL(V),
taking the quotient by the subgroup F*. The interest for algebra is in the process in the other direction: given a projective representation, try to 'lift' it to a conventional linear group representation. This brings in questions of group cohomology.
In fact if one introduces for g in G a lifted element L(g), and a scalar matrix c(g) for g in G representing the freedom in lifting from PGL(V) back to GL(V), and then look at the condition for lifted images to satisfy the homomorphism condition
- L(gh) = L(g)L(h)
after modification by c(g), c(h) and c(gh), one finds a cocycle equation. This need not come down to a coboundary: that is, projective representations may not lift.
It is shown, however, that this leads to an extension problem for G. If G is correctly extended we can speak of a linear representation of the extended group, which gives back the initial projective representation on factoring by F* and the extending subgroup. The solution is always a central extension. From Schur's lemma, it follows that the irreducible representations of central extensions of G, and the projective representations of G, describe essentially the same questions of representation theory
