Projective representation

This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

Definition

Let $G$ be a group. A projective representation of $G$ over a field $k$ is defined in the following equivalent ways:

• It is a homomorphism from $G$ to the projective general linear group for a vector space over $k$
• It is a map $\alpha :G\to GL(V)$ (viz,to the general linear group) where the images of elements of $g$ are ambiguous upto scalar multiples, and such that $\alpha (gh)=\alpha (g)\alpha (h)$ upto a scalar multiple.

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 are termed projectively equivalent if at any $g$, they differ multiplicatively by a scalar matrix.

Facts

Linear representations are 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.

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 porjective representations give 2-cocycles that differ by a 2-coboundary.

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 universal covering group, if such a thing does exist.