Higher Lie characters and cyclic descent extension on conjugacy classes

Ron M. Adin; Pál Hegedüs; Yuval Roichman

doi:10.5802/alco.323

Higher Lie characters and cyclic descent extension on conjugacy classes

Ron M. Adin
, Pál Hegedüs
, Yuval Roichman

Department of Mathematics

Budapest University of Technology and Economics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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 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.

Original language	English
Pages (from-to)	1557-1591
Number of pages	35
Journal	Algebraic Combinatorics
Volume	6
Issue number	6
DOIs	https://doi.org/10.5802/alco.323
State	Published - 2023

Bibliographical note

Publisher Copyright:
© 2023 The Author(s).

Funding

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.

Funders	Funder number
MIT-Israel
hospitality of Bar-Ilan University
Horizon 2020 Framework Programme	741420
European Commission
Bar-Ilan University
Israel Science Foundation	1970/18
Nemzeti Kutatási Fejlesztési és Innovációs Hivatal	K138596

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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 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.",

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AU - Roichman, Yuval

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Higher Lie characters and cyclic descent extension on conjugacy classes

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