It is a result of Benard (Schur indices and splitting fields of the unitary reflection groups, J. Algebra, 1976) that the Schur index over of any irreducible character of a generalized symmetric group is equal to . This post recalls, for the sake of comparison with the literature, other results known about the Schur index in this case.
Suppose that is a finite group and is an irreducible representation of , , for some complex vector space . We say that may be realized over a subfield if there is an vector space and an action of on such that and are equivalent representations of , where acts on by “extending scalars” in from to . Such a representation is called an representation. In other words, is an representation provided it is equivalent to a representation which can be written down explicitly using matrices with entries in .
Suppose that the character of has the property that
for some subfield independent of . It is unfortunately true that, in general, is not necessarily an representation. However, what is remarkable is that, for some , there are representations, , all equivalent to , such that is an representation. The precise theorem is the following remarkable fact.
Theorem: (Schur) Let be an irreducible character and let be any field containing the values of . There is an integer such that is the character of an representation.
The smallest in the above theorem is called the Schur index and denoted .
Next, we introduce some notation:

let denote the extension field of obtained by adjoining all the values of \ ($g\in G$), where is the character of ,

let denote the FrobeniusSchur indicator of (so ),

let denote the Schur multiplier of (by definition, the smallest integer such that $m\chi$ can be realized over (this integer exists, by the abovementioned theorem of Schur).
The following result shows how the Schur index behaves under induction (see Proposition 14.1.8 in G. Karpilovsky,
Group representations, vol. 3, 1994).
Proposition: Let be an irreducible character of and let denote an irreducible character of a subgroup of . If then divides .
A future post shall list some properties of the Schur index in the case where is a generalized symmetric group and is either the reals or rationals.