In general, there are three types of (complex) representations of a finite group . (A good reference for all this is Serre’s well-known book, Linear representations of finite groups.)
Let be an -dimensional irreducible representation of a finite group on a complex vector space . Let denote the character of .
Exactly one of the following possibilities must hold:
- One of the values of the character is not real. Such representations will be called complex (or type 1 or unitary).
- All the values of are real and is realizable by a representation over a real vector space. Such representations will be called real (or type 2 or orthogonal).
- All the values of are real but is not realizable by a representation over a real vector space. Such representations will be called quaternionic (or type 3 or symplectic).
Proposition (Frobenius-Schur): Let be an irreducible representation of a finite group on a complex vector space with character $\chi$. Then
This quantity is sometimes called the Frobenius-Schur indicator of .
It can be shown that if are equivalent representations then and have the same type.
In the next post we will examine the types that the irreducible representations of semi-direct product fall into.