In this post, we discover which representations of the generalized symmetric group can be realized over a given abelian extension of .
Let be the representation defined previously, where .
Let be a subfield, where is a primitive root of unity. Assume contains the field generated by the values of the character of . Assume is Galois and let . Note if we regard as a subset of then there is an induced action of on ,
where , . This action extends to an action on .
In the notation above, if and only if is equivalent to under the action of on .
where and .
Theorem: The character of has values in if and only if ,
for all and all .
This theorem is proven in this paper.
We now determine the splitting field of any irreducible character of a generalized symmetric group.
Theorem: Let be an irreducible character of . We have
This theorem is also proven in this paper.
In the next post we shall give an example.