The main tool for relating real and complex representations is complexification.
Definition 16.2.1 (Complexification). Given a real representation of , its complexification is the complex vector space , with acting by for . This defines a functor .
Definition 16.2.2 (Real complex representation). A complex representation of is called real if there exists a real representation such that .
Proposition 16.2.1 (Necessary condition). If is real, then takes values in . (The trace of a matrix with real entries is real.)
The converse is also true for irreducible representations, as we will show.