Projective representation

In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group

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).^[1]

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

\operatorname{GL}(V,F) \rightarrow \operatorname{PGL}(V, F)

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.

A projective representation of

G

can be pulled back to a linear representation of a central extension

C

of

G.

In general, given a projective representation $ρ : G \to PGL(V)$ it cannot be lifted to a linear representation $G \to GL(V)$ , 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 $GL(V) \to PGL(V)$ 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 $ρ : G \to PGL(V)$ along the quotient map, obtaining a linear representation $σ : C \to GL(V)$ 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.

Group cohomology

The analysis of the lifting question involves group cohomology. Indeed, if one fixes for each $g$ in $G$ a lifted element $L (g)$ in lifting from $PGL(V)$ back to $GL(V)$ , the lifts then satisfy

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

for some scalar $c (g, h)$ in $F *$ . It follows that the 2-cocycle or Schur multiplier $c$ satisfies the cocycle equation

c(h,k)c(g,hk)= c(g,h) c(gh,k)

for all $g, h, k$ in $G$ . This $c$ depends on the choice of the lift $L$ ; a different choice of lift $L' (g) = f (g) L (g)$ will result in a different 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 $H 2 (G, F *)$ . 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.^[2]

In general, a nontrivial class leads to an extension problem for $G$ . If $G$ is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down to $G$ . 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 $G$ , are essentially the same objects.