Editing Projective representation (section)

{{Short description|Map from algebra to geometric transforms}}
In the field of [[representation theory]] in [[mathematics]], a '''projective representation''' of a [[group (mathematics)|group]] ''G'' on a [[vector space]] ''V'' over a [[field (mathematics)|field]] ''F'' is a [[group homomorphism]] from ''G'' to the [[projective linear group]]
<math display="block">\mathrm{PGL}(V) = \mathrm{GL}(V) / F^*,</math>
where GL(''V'') is the [[general linear group]] of invertible [[linear transformation]]s of ''V'' over ''F'', and ''F''<sup>∗</sup> is the [[normal subgroup]] consisting of nonzero scalar multiples of the identity transformation (see [[Scalar transformation]]).<ref>{{Harvnb|Gannon|2006|pp=176–179}}.</ref>

In more concrete terms, a projective representation of <math>G</math> is a collection of operators <math>\rho(g)\in\mathrm{GL}(V),\, g\in G</math> satisfying the homomorphism property up to a constant:
:<math>\rho(g)\rho(h) = c(g, h)\rho(gh),</math>
for some constant <math>c(g, h)\in F</math>. Equivalently, a projective representation of <math>G</math> is a collection of operators <math>\tilde\rho(g)\subset\mathrm{GL}(V), g\in G</math>, such that <math>\tilde\rho(gh)=\tilde\rho(g)\tilde\rho(h)</math>. Note that, in this notation, <math>\tilde\rho(g)</math> is a ''[[Coset|set]]'' of linear operators related by multiplication with some nonzero scalar.

If it is possible to choose a particular representative <math>\rho(g)\in\tilde\rho(g)</math> in each family of operators in such a way that the homomorphism property is satisfied ''exactly'', rather than just up to a constant, then we say that <math>\tilde\rho</math> can be "de-projectivized", or that <math>\tilde\rho</math> can be "lifted to an ordinary representation". More concretely, we thus say that <math>\tilde\rho</math> can be de-projectivized if there are <math>\rho(g)\in\tilde\rho(g)</math> for each <math>g\in G</math> such that <math>\rho(g)\rho(h)=\rho(gh)</math>. This possibility is discussed further below.