# Splitting fields of representations of generalized symmetric groups, 2

In general, there are three types of (complex) representations of a finite group $G$. (A good reference for all this is Serre’s well-known book, Linear representations of finite groups.)

Let $\rho:H\rightarrow Aut(W)$ be an $n$-dimensional irreducible representation of a finite group $G$ on a complex vector space $W$. Let $\chi$ denote the character of $\rho$.

Exactly one of the following possibilities must hold:

• One of the values of the character $\chi$ is not real. Such representations will be called complex (or type 1 or unitary).
• All the values of $\chi$ are real and $\rho$ 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 $\chi$ are real but $\rho$ 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 $\rho:H\rightarrow Aut(W)$ be an irreducible representation of a finite group $G$ on a complex vector space $W$ with character $\chi$. Then

${1\over |G|} \sum_{g\in G}\chi(g^2)= \left\{ \begin{array}{cc} 0,&\rho\ {\rm complex},\\ 1,&\rho\ {\rm real},\\ -1,&\rho\ {\rm quaternionic}. \end{array} \right.$

This quantity is sometimes called the Frobenius-Schur indicator of $\rho$.

It can be shown that if $\rho$ $\rho'\cong \rho$ are equivalent representations then $\rho$ and $\rho'$ have the same type.

In the next post we will examine the types that the irreducible representations of semi-direct product $G=C_\ell^n\, >\!\!\lhd \, S_n$ fall into.