Splitting fields of representations of generalized symmetric groups, 7

In this post, we discover which representations of the generalized symmetric group G = S_n\ wr\ C_\ell = C_\ell^n\, >\!\!\lhd \, S_n can be realized over a given abelian extension of {\mathbb{Q}}.

Let \theta_{\mu,\rho}\in G^* be the representation defined previously, where \rho\in ((S_n)_\mu)^*.

Let K\subset {\mathbb{Q}}(\zeta_\ell) be a subfield, where \zeta_\ell is a primitive \ell^{th} root of unity. Assume K contains the field generated by the values of the character of \theta_{\mu,\rho}. Assume K/{\mathbb{Q}} is Galois and let \Gamma_K=Gal({\mathbb{Q}}(\zeta_\ell)/K). Note if we regard C_\ell as a subset of {\mathbb{Q}}(\zeta_\ell) then there is an induced action of \Gamma_K on C_\ell,

\sigma:\mu \longmapsto \mu^\sigma, \ \ \ \ \ \ \ \ \ \mu\in (C_\ell)^*,\ \ \sigma\in \Gamma_K,

where \mu^\sigma(z)=\mu(\sigma^{-1}(z)), z\in C_\ell. This action extends to an action on (C_\ell^n)^*=(C_\ell^*)^n.

Key Lemma:
In the notation above, \theta_{\mu,\rho}\cong\theta_{\mu,\rho}^\sigma if and only if \mu is equivalent to \mu^\sigma under the action of S_n on (C_\ell^n)^*.

Let

n_\mu(\chi)=|\{i\ |\ 1\leq i\leq n,\ \mu_i=\chi\}|,

where \mu=(\mu_1,...,\mu_n)\in (C_\ell^n)^* and \chi\in C_\ell^*.

Theorem: The character of \theta_{\mu,\rho}\in G^* has values in K if and only if n_\mu(\chi)=n_\mu(\chi^\sigma),
for all \sigma\in \Gamma_K and all \chi\in C_\ell^*.

This theorem is proven in this paper.

We now determine the splitting field of any irreducible character of a generalized symmetric group.

Theorem: Let \chi=tr(\theta_{\rho,\mu}) be an irreducible character of G=S_n\ wr\ C_\ell. We have

Gal({\mathbb{Q}}(\zeta_\ell)/{\mathbb{Q}}(\chi))= Stab_\Gamma(\chi).

This theorem is also proven in this paper.

In the next post we shall give an example.

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