  A now-classical cyclic extension of the descent set of a permutation has been
introduced by Klyachko and Cellini. Following a recent axiomatic approach to
this notion, it is natural to ask which sets of permutations admit such an
extension.
  The main result of this paper is a complete answer in the case of conjucay
classes of permutations. It is shown that the conjugacy class of cycle type
$\lambda$ has such an extension if and only if $\lambda$ is not of the form
$(r^s)$ for some square-free $r$. The proof involves a detailed study of hook
constituents in higher Lie characters.
