Editing Projective representation (section)

===Group cohomology===
The analysis of the lifting question involves [[group cohomology]]. Indeed, if one fixes for each {{math|''g''}} in {{math|''G''}} a lifted element {{math|''L''(''g'')}} in lifting from {{math|PGL(''V'')}} back to {{math|GL(''V'')}}, the lifts then satisfy

:<math>L(gh) = c(g, h)L(g)L(h)</math>

for some scalar {{math|''c''(''g'',''h'')}} in {{math|''F''<sup>∗</sup>}}. It follows that the 2-cocycle or [[Schur multiplier]] {{math|''c''}} satisfies the cocycle equation

:<math> c(h, k)c(g, hk) = c(g, h) c(gh, k)</math>

for all {{math|''g'', ''h'', ''k''}} in {{math|''G''}}. This {{math|''c''}} depends on the choice of the lift {{math|''L''}}; a different choice of lift {{math|''L&prime;''(''g'') {{=}} ''f''(''g'') ''L''(''g'')}} will result in a different cocycle

:<math>c^\prime(g, h) = f(gh)f(g)^{-1} f(h)^{-1} c(g,h)</math>

cohomologous to {{math|''c''}}.  Thus {{math|''L''}} defines a unique class in {{math|H<sup>2</sup>(''G'', ''F''<sup>∗</sup>)}}. This class might not be trivial. For example, in the case of the [[symmetric group]] and [[alternating group]], Schur established that there is exactly one non-trivial class of Schur multiplier, and completely determined all the corresponding irreducible representations.<ref>{{harvnb|Schur|1911}}</ref>

In general, a nontrivial class leads to an [[extension problem]] for {{math|''G''}}. If {{math|''G''}} is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down to {{math|''G''}}. The solution is always a [[Group extension#Central extension|central extension]]. From [[Schur's lemma]], it follows that the [[irreducible representation]]s of central extensions of {{math|''G''}}, and the irreducible projective representations of {{math|''G''}}, are essentially the same objects.