Higher Lie Characters and Cyclic Descent Extension on Conjugacy Classes

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 a (not necessarily classical) extension. The main result of this paper is a complete answer in the case of conjugacy classes of permutations. It is shown that the conjugacy class of cycle type λ has such an extension if and only if λ 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.