Projective representation

In the mathematical field of representation theory, a projective representation of a group "G" on a vector space "V" over a field F is a group homomorphism from "G" to :PGL("V",F) = GL("V",F)/F^∗ where GL("V",F) is the automorphism group of invertible linear transformations of "V" over F and F^* here is the normal subgroup consisting of multiplications of vectors in "V" by nonzero elements of F (that is, scalar multiples of the identity). [Harvnb|Gannon|2006|pp=176–179.]

Linear representations and projective representations

One way in which a projective representation can arise is by taking a linear group representation of "G" on "V" and applying the homomorphism

:GL("V", F) → PGL("V", F),

which is the quotient by the subgroup F^∗. 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".

The analysis of this 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

:"L"("gh") = "c"("g","h")"L"("g")"L"("h")

for some constant "c"("g","h") in F^∗. The 2-cocycle or Schur multiplier "c" must satisfy the cocycle equation

:$c(h,k)c(g,hk)=\; c(g,h)\; c(gh,k)$

for all "g", "h", "k" in "G", A different choice of lift "L' "("g")= "f"("g") "L"("g") will result in a new cocycle

:$c^prime(g,h)\; =\; f(gh)f(g)^\{-1\}\; f(h)^\{-1\}\; c(g,h)$

cohomologous to "c". Thus "L" defines a unique class in H²("G", F^∗), 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. [harvnb|Schur|1911]

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^∗ and the extending subgroup. The solution is always a central extension. From Schur's lemma, it follows that the irreducible representations of central extensions of "G", and the projective representations of "G", describe essentially the same questions of representation theory.

Notes

Reference

*citation|first=I.|last=Schur|authorlink=Issai Schur|title=Über die Dartsellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen|year=1911|journal=Crelle's J.|pages=155-250|volume=139|url=http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=261150
*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=978-0521835312

ee also

*Affine representation
*Group action