Projective representation

In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity; scalar transformations). In more concrete terms, a projective representation is a collection of operators ρ ( g ) , g ∈ G {displaystyle ho (g),,gin G} , where it is understood that each ρ ( g ) {displaystyle ho (g)} is only defined up to multiplication by a constant. These should satisfy the homomorphism property up to a constant: for some constants c ( g , h ) {displaystyle c(g,h)} . Since each ρ ( g ) {displaystyle ho (g)} is only defined up to a constant anyway, it does not strictly speaking make sense to ask whether the constants c ( g , h ) {displaystyle c(g,h)} are equal to 1. Nevertheless, one can ask whether it is possible to choose a particular representative of each family ρ ( g ) {displaystyle ho (g)} of operators in such a way that the ρ ( g ) {displaystyle ho (g)} 's satisfy the homomorphism property on the nose, not just up to a constant. If such a choice is possible, we say that ρ {displaystyle ho } can be 'de-projectivized,' or that ρ {displaystyle ho } can be 'lifted to an ordinary representation.' This possibility is discussed further below. 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). The interest for algebra is in the process in the other direction: given a projective representation, try to 'lift' it to an ordinary linear representation. 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 homology, as described below. However, one can lift a projective representation ρ {displaystyle ho } of G to a linear representation of a different group H, which will be a central extension of G. The group H {displaystyle H} is the subgroup of G × G L ( V ) {displaystyle G imes mathrm {GL} (V)} defined as follows: where π {displaystyle pi } is the quotient map of G L ( V ) {displaystyle mathrm {GL} (V)} onto P G L ( V ) {displaystyle mathrm {PGL} (V)} . Since ρ {displaystyle ho } is a homomorphism, it is easy to check that H {displaystyle H} is, indeed, a subgroup of G × G L ( V ) {displaystyle G imes mathrm {GL} (V)} . If the original projective representation ρ {displaystyle ho } is faithful, then H {displaystyle H} is isomorphic to the preimage in G L ( V ) {displaystyle mathrm {GL} (V)} of ρ ( G ) ⊂ G L ( V ) {displaystyle ho (G)subset mathrm {GL} (V)} .