Higher Lie characters and cyclic descent extension on conjugacy classes

TL;DR

This paper characterizes which conjugacy classes of permutations admit a cyclic descent extension, showing that it depends on the cycle structure, with a complete classification based on the shape of the cycle type.

Contribution

It provides a complete classification of conjugacy classes that admit cyclic descent extensions, connecting the problem to higher Lie characters and hook constituents.

Findings

Conjugacy class of cycle type λ admits extension iff λ is not of form (r^s) with r square-free.

The proof involves analyzing hook constituents in higher Lie characters.

The result generalizes previous notions of cyclic descent extensions to conjugacy classes.

Abstract

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.