Projective representation

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This term is related to: linear representation theory
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Let G be a group. A projective representation of G over a field k is defined in the following equivalent ways:

if we let f: G \times G \to k^* be the function such that:

\alpha(gh) = f(g,h)\alpha(g)\alpha(h)

then we say that \alpha is a f-representation.

Two projective representations \alpha_1: G \to GL(V_1) and \alpha_2:G \to GL(V_2) over a field k are termed projectively equivalent if there exists a vector space isomorphism F:V_1 \to V_2 and a function (not necessarily a homomorphism) \theta:G \to k^\times such that for every g \in G and v\in V:

F(\alpha_1(g) \cdot v) = \theta(g)(\alpha_2(g) \cdot F(v))

In other words, they differ by a scalar multiplication combined with a change-of-basis isomorphism.


Linear representations give projective representations

Every linear representation G \to GL(V) gives rise to a projective representation, G \to PGL(V), simply by composing the given representation with the quotient map GL(V) \to PGL(V) (which involves quotienting out by the center). However, not every projective representation arises from a linear representation.

However, it is very much possible that different linear representations descend to the same projective representation. The following is in fact true:

Two linear representations are projectively equivalent if and only if one of them can be obtained from the other via multiplication by a one-dimensional representation.

In particular, all the one-dimensional representations are projectively equivalent to each other.

Projective representation gives a 2-cocycle

Let \alpha be a projective representation. Then we can associate to it a 2-cocycle such that:

\alpha(gh) = f(g,h)\alpha(g)\alpha(h)

By the assumptions for a projective representation, this turns out to be a 2-cocycle from G to k^*.

It turns out that projectively equivalent projective representations give 2-cocycles that differ multiplicatively by a 2-coboundary. Thus, any projective representation up to projective equivalence defines an element of the second cohomology group for trivial group action H^2(G,k^\ast).

When is a projective representation equivalent to a linear representation?

A projective representation is projectively equivalent to a linear representation iff the 2-cocycle associated to it is a 2-coboundary. In particular, this means that if H^2(G,k^*) (the second cohomology group) is trivial, any projective representation is projectively equivalent to a linear representation.

When k = \mathbb{C}, this is the same as the assertion that the group has trivial Schur multiplier (or is Schur-trivial).

In general, any projective representation of the group gives rise to a linear representation of its Schur covering group.