# Splitting fields of representations of generalized symmetric groups, 1

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 $C_\ell$ denote the cyclic group of order $\ell\geq 1$, let $S_n$ denote the symmetric group of degree $n\geq 1$, and let $G$ denote the semi-direct product $G=C_\ell^n\, >\!\!\lhd \, S_n$. We think of this as the set of pairs $(v,p)$, with

• $v=(v_1,...,v_n)$, where each $v_i\in C_\ell=\{0,1,...,\ell-1\}$,
• $p\in S_n$,
• $S_n$ acts on $C_\ell^n$ by $p(v)=(v_{p(1)},v_{p(2)},...,v_{p(n)})$,
• multiplication given by $(v,p)*(v',p')=(v+p(v'),pp'),\ \ \ \ \ (v,p),(v',p')\in G.$

A group of the form $C_\ell^n\, >\!\!\lhd \, S_n$, also written $S_n \ {\rm wr}\ C_\ell$ (here wr denotes the wreath product), will be called a generalized symmetric group.

We may identify each element of $C_\ell^n\, >\!\!\lhd \, S_n$ with an $n\times n$ monomial matrix (a matrix with exactly one non-zero entry in each row and column) with entries in $C_\ell$.

Our main goal will be to investigate the following question: What is the smallest extension of ${\mathbb{Q}}$ required to realize (using matrices) a given irreducible character of a generalized symmetric group?