A generalisation of bar-core partitions

Yates, Dean

doi:10.5802/alco.231

Yates, Dean ¹

¹ Queen Mary, University of London Mile End Road Bethnal Green London E1 4NS

Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 667-698.

Abstract

When $p$ and $q$ are coprime odd integers no less than 3, Olsson proved that the $q$ -bar-core of a $p$ -bar-core is again a $p$ -bar-core. We establish a generalisation of this theorem: that the $p$ -bar-weight of the $q$ -bar-core of a bar partition $λ$ is at most the $p$ -bar-weight of $λ$ . We go on to study the set of bar partitions for which equality holds and show that it is a union of orbits for an action of a Coxeter group of type ${\tilde{C}}_{(p - 1) / 2} \times {\tilde{C}}_{(q - 1) / 2}$ . We also provide an algorithm for constructing a bar partition in this set with a given $p$ -bar-core and $q$ -bar-core.

Received: 2021-07-08
Revised: 2022-02-09
Accepted: 2022-02-13
Published online: 2022-09-08

DOI: 10.5802/alco.231

Classification: 05E10, 05A17, 20C30
Keywords: representation theory, symmetric group, partitions, projective, bar-core

Author's affiliations:

Yates, Dean ¹

¹ Queen Mary, University of London Mile End Road Bethnal Green London E1 4NS

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Yates, Dean. A generalisation of bar-core partitions. Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 667-698. doi : 10.5802/alco.231. https://alco.centre-mersenne.org/articles/10.5802/alco.231/

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