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| In the [[mathematics|mathematical]] field of [[representation theory]], 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]]
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| :PGL(''V'',''F'') = GL(''V'',''F'')/''F''<sup>∗</sup> | |
| where GL(''V'',''F'') is the [[general linear group]] of invertible linear transformations of ''V'' over ''F'' and ''F''<sup>*</sup> here is the [[normal subgroup]] consisting of multiplications of vectors in ''V'' by nonzero elements of ''F'' (that is, scalar multiples of the identity; [[scalar transformation]]s).<ref>{{Harvnb|Gannon|2006|pp=176–179}}.</ref>
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| ==Linear representations and projective representations==
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| 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
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| :<math>GL(V,F) \rightarrow PGL(V, F)</math>
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| which is the quotient by the subgroup ''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 a conventional ''linear representation''.
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| [[File:Projective-representation-lifting.svg|225px|thumb|A projective representation of ''G'' can be pulled back to a linear representation of a central extension ''C'' of ''G.'']]
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| In general, given a projective representation {{math|''ρ'': ''G'' → PGL(''V'')}} it cannot be lifted to a linear representation {{math|''G'' → GL(''V'')}}, and the [[obstruction theory|obstruction]] to this lifting can be understood via group homology, as described below. However, one ''can'' lift a projective representation of ''G'' to a linear representation of a different group ''C,'' which will be a [[central extension (mathematics)|central extension]] of ''G.'' To understand this, note that GL(''V'') → PGL(''V'') is a central extension of PGL, meaning that the kernel is central (in fact, is exactly the center of GL). One can [[pullback|pull back]] the projective representation {{math|''ρ'': ''G'' → PGL(''V'')}} along the quotient map, obtaining a ''linear'' representation {{math|''σ'': ''C'' → GL(''V'')}} and ''C'' will be a central extension of ''G'' because it is a pullback of a central extension. Thus projective representations of ''G'' can be understood in terms of linear representations of (certain) central extensions of ''G.'' Notably, for ''G'' a [[perfect group]] there is a single [[universal perfect central extension]] of ''G'' that can be used.
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| ===Group cohomology===
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| The analysis of the lifting question involves [[group cohomology]]. Indeed, if one introduces for ''g'' in ''G'' a lifted element ''L''(''g'') in lifting from PGL(''V'') back to GL(''V''), the lifts must satisfy
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| :<math>L(gh) = c(g,h)L(g)L(h)</math> | |
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| for some scalar ''c''(''g'',''h'') in ''F''<sup>∗</sup>. The 2-cocycle or [[Schur multiplier]] ''c'' must satisfy the cocycle equation
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| :<math> c(h,k)c(g,hk)= c(g,h) c(gh,k)</math> | |
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| for all ''g'', ''h'', ''k'' in ''G''. This ''c'' depends on the choice of the lift ''L'', but a different choice of lift ''L′''(''g'')= ''f''(''g'') ''L''(''g'') will result in a new cocycle
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| :<math>c^\prime(g,h) = f(gh)f(g)^{-1} f(h)^{-1} c(g,h)</math>
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| cohomologous to ''c''. Thus ''L'' defines a unique class in H<sup>2</sup>(''G'', ''F''<sup>∗</sup>), which need not be trivial. For example, in the case of the [[symmetric group]] and [[alternating group]], Schur proved that there is exactly one non-trivial class of Schur multiplier and completely determined all the corresponding irreducible representations.<ref>{{harvnb|Schur|1911}}</ref>
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| 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 ''F''<sup>∗</sup> and the extending subgroup. 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 ''G'', and the irreducible projective representations of ''G'', describe essentially the same questions of representation theory.
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| ==Projective representations of Lie groups==
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| {{see also|Spinor}}
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| Studying projective representations of [[Lie group]]s leads one to consider true representations of their central extensions (see [[Group extension#Lie groups]]). In many cases of interest it suffices to consider representations of [[covering group]]s; for a connected Lie group ''G'', this amounts to studying the representations of the Lie algebra of ''G''. Notable cases of covering groups giving interesting projective representations:
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| * The [[special orthogonal group]] SO(''n'',''F'') is doubly covered by the [[Spin group]] Spin(''n'',''F''). In particular, the [[rotation group SO(3)|group SO(3,'''R''')]] (the rotation group in 3 dimensions) is doubly covered by [[special unitary group|SU(2)]]. This has important applications in quantum mechanics, as the [[representation theory of SU(2)|study of representations of SU(2)]] leads to a [[Galilean relativity|low-velocity]] theory of [[spin (physics)|spin]].
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| * The group [[Lorentz group|SO<sup>+</sup>(3;1)]], isomorphic to the [[Möbius group]], is likewise doubly covered by [[special linear group|SL<sub>2</sub>]]('''C'''). Both are supergroups of aforementioned SO(3) and SU(2) respectively and form a [[special relativity|relativistic]] spin theory.
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| * The [[orthogonal group]] O(''n'') is double covered by the [[Pin group]] Pin<sub>±</sub>(''n'').
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| * The [[symplectic group]] Sp(2''n'') is double covered by the [[metaplectic group]] Mp(2''n'').
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|first=I.|last=Schur|authorlink=Issai Schur|title=Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen|year=1911|journal=[[Crelle's Journal]]|pages=155–250|volume=139|url=http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=261150
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| }}
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| *{{citation|title=Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics|first=Terry|last= Gannon|publisher=Cambridge University Press|year= 2006|isbn=978-0-521-83531-2}}
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| ==See also==
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| *[[Affine representation]]
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| *[[Group action]]
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| [[Category:Homological algebra]]
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| [[Category:Group theory]]
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| [[Category:Representation theory]]
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| [[Category:Representation theory of groups]]
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