Projective representation
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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).[1]
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[edit] 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
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.[2]
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.
[edit] Notes
- ^ Gannon 2006, pp. 176–179.
- ^ Schur 1911
[edit] References
- Schur, I. (1911), "Über die Dartsellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen", Crelle's J. 139: 155–250, http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=261150
- Gannon, Terry (2006), Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, ISBN 978-0521835312
