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.

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