Higher Lie characters and cyclic descent extension on conjugacy classes

Ron M. Adin, Pál Hegedüs, Yuval Roichman

Research output: Contribution to journalArticlepeer-review


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 (rs) for some square-free r. The proof involves a detailed study of hook constituents in higher Lie characters.

Original languageEnglish
Pages (from-to)1557-1591
Number of pages35
JournalAlgebraic Combinatorics
Issue number6
StatePublished - 2023

Bibliographical note

Publisher Copyright:
© 2023 The Author(s).


The second author is grateful for the hospitality of Bar-Ilan University during his Sabbatical leave. Special thanks are due to Jan Saxl, whose personality influenced the development of this paper in many ways; in particular, the paper is inspired by the ingenious use of a higher Lie character in [18]. Thanks also to Eli Bagno, Jonathan Bloom, Sergi Elizalde, Darij Grinberg, Vic Reiner, Richard Stanley, Sheila Sundaram and Josh Swanson for their comments. Acknowledgements. PH was partially supported by Hungarian National Research, Development and Innovation Office (NKFIH) Grant No. K138596 and by Bar-Ilan University visiting grant. The project leading to this application has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant agreement No. 741420. RMA and YR were partially supported by the Israel Science Foundation, grant no. 1970/18 and by an MIT-Israel MISTI grant.

FundersFunder number
hospitality of Bar-Ilan University
Horizon 2020 Framework Programme741420
European Research Council
Bar-Ilan University
Israel Science Foundation1970/18
Nemzeti Kutatási Fejlesztési és Innovációs HivatalK138596


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