Projective representation

In mathematics, in particular in group theory, if G is a group and ρ is a vector space over a field K, then a projective representation is a homomorphism from G to Aut(ρ)/Kx where Kx here is the normal subgroup of Aut(ρ) consisting of multiplications of vectors in ρ by nonzero elements of K (that is, scalar multiples of the identity), and Aut(ρ) means the automorphism group of the vector space underlying ρ. This may be 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) to PGL(V), taking the quotient by the subgroup Kx. 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 looks 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 Kx and the extending subgroup.

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