# Getting new irreducible representations from old ones

• A
Suppose I had some group G, and I could classify all of its irreducible K-representations for some K = R,C, or H. Given that information (how) can I classify its irreducible K-representations for all K.

i.e. suppose I knew all the irreducible real representations of G, (how) could I then get all the irreducible complex and quaternionic representations?

fresh_42
Mentor
A standard method in case of linear representations on a real vectorspace ##V_\mathbb{R}## is to construct ##V_\mathbb{C} = V_\mathbb{R} \otimes_\mathbb{R} \mathbb{C}## and define for a given ##\varphi_\mathbb{R} \, : \,G \longrightarrow GL(V_\mathbb{R})##
$$\varphi_\mathbb{C} \, : \,G \longrightarrow GL(V_\mathbb{C}) \quad \text{ by } \quad \varphi_\mathbb{C}(g)(\lambda\cdot v) = \varphi_\mathbb{R}(g)(v) \otimes \lambda$$
I'm not sure, however, whether they automatically will be irreducible again, as there are simply more eigenvalues available, so I doubt it.

Infrared
Indeed, let the cyclic group of order 3 act on $\mathbb{R}^2$ with a generator corresponding to rotation by $2\pi/3$. This representation is irreducible but its complexification isn't.