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?

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