Enumeration of orbits on cycles for linear and affine groups

Abstract

In this paper we compute the Parker vectors of the general and special linear groups, and of the affine groups, considered as permutation groups on the set of the vectors on which they act. The Parker vector of a permutation group, as we recall in the first section, is a sequence of natural numbers enumerating the orbits of the actions of G on the set of cycles appearing in its elements. In giving an explicit and constructive way of computing these Parker vectors, we get some insight on the cycle structure of the elements of these groups, and on possible representatives for their conjugacy classes.