The main result of this series of blog posts (originally written 1998), which is expository, is to determine (in a sense made precise later) the splitting field of any irreducible character of a generalized symmetric group. This was basically solved by M. Benard in a 1976 J. of Algebra paper. We use one of his results to make the splitting field explicit.
Notation and definitions
Let denote the cyclic group of order , let denote the symmetric group of degree , and let denote the semi-direct product . We think of this as the set of pairs , with
- , where each ,
- acts on by ,
- multiplication given by
A group of the form , also written (here wr denotes the wreath product), will be called a generalized symmetric group.
We may identify each element of with an monomial matrix (a matrix with exactly one non-zero entry in each row and column) with entries in .
Our main goal will be to investigate the following question: What is the smallest extension of required to realize (using matrices) a given irreducible character of a generalized symmetric group?