Irreducible projective representation

Definition

A projective representation is termed irreducible if the vector space being acted upon has dimension at least one, and there is no proper nonzero subspace of the vector space that is invariant under the image of the representation. Note that a projective general linear group has a well defined action on vector subspaces, even though its action on individual vectors is not well-defined.

Facts

• A linear representation is an irreducible linear representation if and only if the corresponding projective representation is an irreducible projective representation.
• Given a group $G$ and a normal subgroup $H$, a projective representation of $G/H$ is irreducible if and only if the corresponding (composite) projective representation of $G$ is irreducible.