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 $\theta_{\mu,\rho}$ .

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

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Yet Another Mathblog

Splitting fields of representations of generalized symmetric groups, 6

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Yet Another Mathblog

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