Characters, fields, Schur indices and divisibility

摘要

Let G be a finite nilpotent group. Suppose that G_0 is a subgroup of G and that ψ is an irreducible character of G_0. Consider the set S whose elements are the natural numbers m_Q[Q(χ):Q], as χ runs through the irreducible characters of G which contain ψ as a summand when restricted to G _0. Here m_Q(χ) is, as usual, the rational Schur index of χ, and [Q(χ): 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.