Editing Projective representation (section)

==Linear representations and projective representations==

One way in which a projective representation can arise is by taking a linear [[group representation]] of {{math|''G''}} on {{math|''V''}} and applying the quotient map

:<math>\operatorname{GL}(V, F) \rightarrow \operatorname{PGL}(V, F)</math>

which is the quotient by the subgroup {{math|''F''<sup>∗</sup>}} of [[scalar transformation]]s ([[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 {{math|''ρ'': ''G'' → PGL(''V'')}} cannot be lifted to a linear representation {{math|''G'' → GL(''V'')}}, and the [[obstruction theory|obstruction]] to this lifting can be understood via [[group cohomology]], as described below.

However, one ''can'' lift a projective representation <math>\rho</math> of {{math|''G''}} to a linear representation of a different group {{math|''H''}}, which will be a [[central extension (mathematics)|central extension]] of {{math|''G''}}. The group <math>H</math> is the subgroup of <math>G\times\mathrm{GL}(V)</math> defined as follows:
:<math>H = \{(g, A) \in G\times\mathrm{GL}(V) \mid \pi(A) = \rho(g)\}</math>,
where <math>\pi</math> is the quotient map of <math>\mathrm{GL}(V)</math> onto <math>\mathrm{PGL}(V)</math>. Since <math>\rho</math> is a homomorphism, it is easy to check that <math>H</math> is, indeed, a subgroup of <math>G\times\mathrm{GL}(V)</math>. If the original projective representation <math>\rho</math> is faithful, then <math>H</math> is isomorphic to the preimage in <math>\mathrm{GL}(V)</math> of <math>\rho(G)\subseteq\mathrm{PGL}(V)</math>.

We can define a homomorphism <math>\phi:H\rightarrow G</math> by setting <math>\phi((g, A)) = g</math>. The kernel of <math>\phi</math> is:
:<math>\mathrm{ker}(\phi) = \{(e, cI) \mid c \in F^*\}</math>,
which is contained in the center of <math>H</math>. It is clear also that <math>\phi</math> is surjective, so that <math>H</math> is a central extension of <math>G</math>. We can also define an ordinary representation <math>\sigma</math> of <math>H</math> by setting <math>\sigma((g, A)) = A</math>. The ''ordinary'' representation <math>\sigma</math> of <math>H</math> is a lift of the ''projective'' representation <math>\rho</math> of <math>G</math> in the sense that:
:<math>\pi(\sigma((g, A))) = \rho(g) = \rho(\phi((g, A)))</math>.

If {{math|''G''}} is a [[perfect group]] there is a single [[universal perfect central extension]] of {{math|''G''}} that can be used.

===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.