Character fields and Clifford's theorem

Question

Let $G$ be a finite group with normal subgroup $N$. Let $\chi$ be an irreducible complex character of $G$. Then Clifford's Theorem says that $\mathrm{res}^{G}_{N}\chi = e(\eta_1 + \cdots + \eta_r)$ where $e$ is a positive integer and each $\eta_i$ is a complex irreducible character of $N$ such that the $\eta_i$'s are the distinct conjugates of $\eta := \eta_1$ (i.e. for each $i$ there exists $g_i \in G$ such that $\eta_i(n)=\eta(g_i n g_i^{-1})$ for all $n \in N$.) Let $\mathbb{Q}(\eta) = \mathbb{Q}(\eta(n) : n \in N)$ be the character field of $\eta$ and similarly for $\mathbb{Q}(\mathrm{res}^{G}_{N} \chi)$. Since $\mathbb{Q}(\eta_i)=\mathbb{Q}(\eta)$ for each $i$, we have $K:=\mathbb{Q}(\mathrm{res}^{G}_{N} \chi) \subseteq \mathbb{Q}(\eta)$.

Question: Is $e$ somehow related to $[\mathbb{Q}(\eta):K]$? For example, do we have $e \geq [\mathbb{Q}(\eta):K]$? Otherwise, can you tell me anything about the relation between $K$ and $\mathbb{Q}(\eta)$? e.g. do we somehow get $\mathbb{Q}(\eta)$ from $K$ by adjoining certain roots of unity? If so, which ones (are they related to say $[G:N]$)? (I am actually interested in the case where we replace $\mathbb{Q}$ by the $p$-adics $\mathbb{Q}_p$ throughout and $[G:N]$ is a power of $p$.)

Answer 1

I don't think there's any clear cut relationship. In particular, the inequality $e \ge [\mathbb Q(\eta) : K]$ needn't hold. For example, take $G=S_5$, $N=A_5$ and let $\chi$ be the unique irreducible $S_5$-character of degree $6$. Then we have $\mathrm{res}^{G}_{N} \chi = \eta + \bar{\eta}$, where $\eta$ and $\bar{\eta}$ are the degree $3$ irreducible characters of $A_5$. So here $e=1$ and $\mathbb Q(\eta) = \mathbb Q(\sqrt 5)$. On the other hand, $\chi$ is integer valued so $K=\mathbb Q$. Thus $e < [\mathbb Q(\eta) : K]$ in this case.

Answer 2

The coefficient $e$ is always $1$ if $G/N$ is cyclic, but $|Q(\eta):Q(\chi_N)|$ can be arbitrarily large. For example, let $N$ be cyclic of prime order $p$, and let $G$ be the semidirect product of a cyclic group of order $p-1$ acting faithfully on $N$. Then $G$ has a rational valued character $\chi$ of degree $p-1$ and the field $Q(\eta)$ is the cyclotomic field of $p$th roots of unity. Then $|Q(\eta):Q(\chi_N)| = |Q(\eta):Q| = p-1$.

It is also true that $e$ can be much larger than $|Q(\eta):Q(\chi_N)|$. For example, take $G$ to be extraspecial of order $2^{2n+1}$ and $N = Z(G)$. Here $e = 2^n$ and both $Q(\eta)$ and $Q(\chi_N)$ are just the rationals $Q$.

Answer 3

A general technique for handling such problems is the following: Let $T=G_{\eta}$ be the inertia subgroup of $\eta$ and $\hat{T}$ the inertia group of its Galois orbit, that is, $\newcommand{\QQ}{\mathbb{Q}} \DeclareMathOperator{\Gal}{Gal}$ $$ T = \{g\in G \mid \eta^g = \eta \} \quad \text{and}\quad \hat{T} = \{ g\in G \mid \exists \alpha_g \in \Gal(\QQ(\eta)/\QQ ) \colon \eta^{g\alpha_g}=\eta \}.$$ By Clifford theory, there is a unique character $\psi$ of $T$ such that $\chi=\psi^G$ and $\psi$ lies over $\eta$. Set $\hat{\psi}:= \psi^{\hat{T}}$. Then the following hold:

1. $\QQ(\chi)=\QQ(\hat{\psi})$ and $\QQ(\psi) = \QQ(\chi, \eta)$.

2. The map $\hat{T}\ni t \mapsto \alpha_t\in \Gal( \QQ(\eta)/ \QQ( \chi_N) )$ is a group homomorphism with kernel $T$ (in general not surjective). In particular, $T$ is normal in $\hat{T}$ with abelian factor group.

3. The subgroup $\Gal( \QQ(\eta) / \QQ(\eta) \cap \QQ(\chi) )$ is contained in the image of $\hat{T}\ni t \mapsto \alpha_t$. (Notice that this Galois group is isomorphic to $\Gal( \QQ(\psi) / \QQ(\chi) )$ by 1.)

This can be shown by elementary Clifford and Galois theory ( Theorem 1 in: Udo Riese and Peter Schmid, MR 1388863 Schur indices and Schur groups. II, J. Algebra 182 (1996), no. 1, 183--200. )

It has been pointed out in the other answers that the relation between the $e$ and the fields is essentially arbitrary. The extension $\QQ(\eta)/\QQ(\chi_N)$ is also arbitrary, except that both fields must be contained in some cyclotomic extension of $\QQ$: Suppose that $\QQ \subseteq \mathbb{F}\subseteq \mathbb{E} \subseteq \QQ(\epsilon)$, where $\epsilon$ is a primitive $n$-th root of unity. Let $G$ be the semidirect product of $\Gal( \QQ(\epsilon) / \mathbb{F} )$ with $C_n =\langle \epsilon\rangle$, and $N$ the semidirect product of $\Gal( \QQ(\epsilon) /\mathbb{E} )$ with $C_n$. Then every faithful linear character of $C_n$ induces to an irreducible character $\chi$ of $G$ with $\QQ(\chi_N)=\mathbb{F}=\QQ(\chi)$ and $\QQ(\eta) =\mathbb{E}$.