Projective representation - Definition and Overview

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)/K^x where K^x 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 K^x. 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 K^x and the extending subgroup.

See also linear representation, affine representation, group action.

Example Usage of representation