- PGL(V, F) = GL(V, F) / F∗,
where GL(V, F) is the general linear group of invertible linear transformations of V over F and F∗ is the normal subgroup consisting of multiplications of vectors in V by nonzero elements of F (that is, scalar multiples of the identity; scalar transformations).
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 quotient map
which is the quotient by the subgroup F∗ of scalar transformations (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.
In general, given a projective representation ρ: G → PGL(V) it cannot be lifted to a linear representation G → GL(V), and the 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 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 pull back the projective representation ρ: G → PGL(V) along the quotient map, obtaining a linear representation 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.
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
for some scalar c(g,h) in F∗. The 2-cocycle or Schur multiplier c must satisfy the cocycle equation
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
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.
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 irreducible projective representations of G, describe essentially the same questions of representation theory.
Projective representations of Lie groups
Studying projective representations of Lie groups 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 groups; 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:
- The special orthogonal group SO(n,F) is doubly covered by the Spin group Spin(n,F). In particular, the group SO(3,R) (the rotation group in 3 dimensions) is doubly covered by SU(2). This has important applications in quantum mechanics, as the study of representations of SU(2) leads to a low-velocity theory of spin.
- The group SO+(3;1), isomorphic to the Möbius group, is likewise doubly covered by SL2(C). Both are supergroups of aforementioned SO(3) and SU(2) respectively and form a relativistic spin theory.
- The orthogonal group O(n) is double covered by the Pin group Pin±(n).
- The symplectic group Sp(2n) is double covered by the metaplectic group Mp(2n).
- Schur, I. (1911), "Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen", Crelle's Journal 139: 155–250
- Gannon, Terry (2006), Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, ISBN 978-0-521-83531-2