Characters, fields, Schur indices and divisibility

Abstract

Let G be a finite nilpotent group. Suppose that G ₀ is a subgroup of G and that ${\psi}$ is an irreducible character of G ₀. Consider the set S whose elements are the natural numbers

$${\rm m}_{\bf Q}(\chi)[{\bf Q}(\chi) : {\bf Q}]$$

as ${\chi}$ runs through the irreducible characters of G which contain ${\psi}$ as a summand when restricted to G ₀. Here m _Q (χ) is, as usual, the rational Schur index of ${\chi}$ , and ${[{\bf Q}(\chi) : {\bf Q}]}$ is the degree of the extension of the field of values of the character as an extension of the rationals. We prove that then the minimum element of S divides all the other elements of S. The result is not true when G is an arbitrary finite group. We also consider some variations of this result.