We have studied actions of finite groups on complex vector spaces. Now we consider the real case and the relationship between real and complex representations.
Let be a finite group. A real representation of is a real vector space together with a group homomorphism (automorphisms as an -vector space).
The averaging trick works over : starting from any inner product on and averaging over , we obtain a -invariant (positive-definite, symmetric) inner product. So if is a -invariant subspace, then is also -invariant, and every real representation of a finite group is completely reducible.
However, Schur’s lemma in its strong form fails over . For instance, acts on by rotation by , which is irreducible (for ) but has nontrivial endomorphisms: any rotation commuting with rotation by works, so (not just ).