Characterizing Real Representations — Studies in Algebra and Group Theory

16.3 Characterizing Real Representations

Proposition 16.3.1 (Antilinear involution criterion). A complex representation of is real if and only if there exists a -equivariant, complex antilinear map (i.e. for all ) with .

Proof. () If , define for (complex conjugation with respect to the real form ). Then is antilinear, , and -equivariant since preserves .

() Given such a , every decomposes as where is the -eigenspace and is the -eigenspace. Since is antilinear, , so . Thus is an -subspace (not a -subspace!) with . Since is -equivariant, is preserved by , giving a real representation with . ∎