projective representation
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
See also
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