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 Frobenius-Schur 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 above-mentioned 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.