# Splitting fields of representations of generalized symmetric groups, 6

This post shall list some properties of the Schur index $m_F(G)$ in the case where $G = S_n\ wr\ C_\ell$ is a generalized symmetric group and $F$ is either the reals or rationals.

Let $\eta_k(z)=z^k$, for $z\in C_\ell$, $1\leq k\leq \ell$.

Theorem: Let $G = S_n\ wr\ C_\ell$. Let $\mu=(\eta_{e_1},...,\eta_{e_n})\in (C_\ell^n)^*$, for some $e_j\in \{0,...,\ell-1\}$, and let $\rho\in (S_n)_\mu^*$. Let
$\chi$ denotes the character of $\theta_{\mu,\rho}$.

1. Suppose that one of the following conditions holds:
1. $4|\ell$ and $\overline{e_1+...+e_n}$ divides $\overline{\ell/4}$ in ${\mathbb{Z}}/\ell {\mathbb{Z}}$, or
2. $(e_1+...+e_n,\ell)=1$,

Then $m_{\Bbb{Q}}(\chi)=1$.

2. Suppose that one of the following conditions holds:
1. $(n,\ell)=1$, $4|\ell$, and $(e_1+...+e_n)x\equiv \ell /4\ ({\rm mod}\ \ell)$ is not solvable, or
2. $(n,\ell)=1$ and $(e_1+...+e_n,\ell)>1$.

Then $m_{\mathbb{Q}}(\chi\eta_1)=1$.

This theorem is proven in this paper. Benard has shown that $m_{\mathbb{Q}}(\chi)=1$, for all $\chi$ as in the above theorem.

Since the Schur index over ${\mathbb{Q}}$ of any irreducible character $\chi$ of a generalized symmetric group $G$ is equal to $1$, each such character is associated to a representation $\pi$ all of whose matrix coefficients belong to the splitting field ${\mathbb{Q}}(\chi)$.

What is the splitting field ${\mathbb{Q}}(\chi)$, for $\chi\in G^*$?

This will be addressed in the next post.