Characters, fields, Schur indices and divisibility. (Q387609)

Let \(G\) be a finite group and let \(\chi\in\mathrm{Irr}(G)\). Let \(\mathbb Q(\chi)\) denote the field generated by the values of \(\chi\), and let \(K\) be a minimal extension of \(\mathbb Q(\chi)\) such that there is a \(KG\)-module affording \(\chi\). Then \(|K:\mathbb Q(\chi)|=m_{\mathbb Q}(\chi)\) is the \textit{Schur index} of the character \(\chi\). The paper under review concerns a divisibility property of the numbers \(|\mathbb Q(\chi):\mathbb Q|\) and \(m_{\mathbb Q}(\chi)|\mathbb Q(\chi):\mathbb Q|\). The main theorem states that if \(G\) is nilpotent, \(G_0\) is a subgroup of \(G\), \(\psi\in\mathrm{Irr}(G_0)\) and \(\mathrm{Irr}(G|_\psi)=\{\chi\in\mathrm{Irr}(G):\langle\chi\!\downarrow_{G_0},\psi\rangle\geq 1\}\) then the least element of \[ S=\{m_{\mathbb Q}(\chi)|\mathbb Q(\chi):\mathbb Q|:\chi\in\mathrm{Irr}(G|_\psi)\} \] divides all other elements of \(S\). The same result holds with \(m_{\mathbb Q}(\chi)|\mathbb Q(\chi):\mathbb Q|\) replaced with \(|\mathbb Q(\chi):\mathbb Q|\). A similar result is proved when \(G\) is an arbitrary finite group under the assumptions that \(G'\leq G_0\leq G\) and \(\psi\in\mathrm{Irr}(G_0)\) extends to a character of its inertial group. -- An example is given to show that the main theorem does not hold (in general) if \(G\) is soluble but not nilpotent. The paper is motivated by a potential application to work of \textit{T. Dokchitser} and \textit{A. Bartel}, [``Rational representations and permutation representations of finite groups'', \url{arXiv 1405.6616}] on the characters of a finite group \(G\) that can be obtained as a \(\mathbb Z\)-linear combination of permutation characters of \(G\).