So far we have studied actions of finite groups on complex vector spaces. In this chapter we consider actions on real vector spaces and how they sit among the complex representations. We will see that an irreducible complex representation falls into exactly one of three classes: it comes from a real representation, it carries a quaternionic structure, or its character is not real-valued. A number computed from the character, the Frobenius–Schur indicator, tells which.
Throughout, is a finite group and all representations are finite-dimensional.
Definition 16.1.1 (Representation over ). A representation of over is a finite-dimensional real vector space together with a group homomorphism , where is the group of invertible -linear maps .
The averaging argument of Lemma 13.2.1 works over . If is any inner product on , then is a -invariant inner product. If is a -invariant subspace, then so is its orthogonal complement , exactly as in the proof of Maschke’s theorem (Theorem 13.2.2). Hence every representation of a finite group over is a direct sum of irreducible ones.
Part (i) of Schur’s lemma (Lemma 13.3.1) also holds over , with the same proof: a nonzero equivariant map between irreducible representations is an isomorphism. Part (ii) fails, because its proof needs an eigenvalue.
Example 16.1.2 (Rotations of the plane). Let , and let act on by letting act by the rotation Since , the rotation has no real eigenvalue, so no line is invariant and the representation is irreducible. A direct computation shows that the matrices commuting with are those of the form , so which is larger than .