Higher Lie characters and cyclic descent extension on conjugacy classes

Adin, Ron M.; Hegedüs, Pál; Roichman, Yuval

doi:10.5802/alco.323

Adin, Ron M.¹; Hegedüs, Pál ²; Roichman, Yuval ¹

¹ Department of Mathematics Bar-Ilan University Ramat-Gan 52900 Israel
² Department of Algebra Institute of Mathematics Budapest University of Technology and Economics Műegyetem rkp. 3 H-1111 Budapest Hungary

Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1557-1591.

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.

Received: 2019-11-13
Revised: 2023-02-10
Accepted: 2023-06-30
Published online: 2024-01-08

DOI: 10.5802/alco.323

Classification: 05E10, 05E05, 20B30, 20C30
Keywords: Cyclic descent, conjugacy class, symmetric group, higher Lie character, hook constituent

Author's affiliations:

Adin, Ron M. ¹; Hegedüs, Pál ²; Roichman, Yuval ¹

¹ Department of Mathematics Bar-Ilan University Ramat-Gan 52900 Israel
² Department of Algebra Institute of Mathematics Budapest University of Technology and Economics Műegyetem rkp. 3 H-1111 Budapest Hungary

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Higher {Lie} characters and cyclic descent extension on conjugacy classes},
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Adin, Ron M.; Hegedüs, Pál; Roichman, Yuval. Higher Lie characters and cyclic descent extension on conjugacy classes. Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1557-1591. doi : 10.5802/alco.323. https://alco.centre-mersenne.org/articles/10.5802/alco.323/

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