# ALGEBRAIC COMBINATORICS

A generalisation of bar-core partitions
Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 667-698.

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 $\lambda$ is at most the $p$-bar-weight of $\lambda$. 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 ${\stackrel{˜}{C}}_{\left(p-1\right)/2}×{\stackrel{˜}{C}}_{\left(q-1\right)/2}$. We also provide an algorithm for constructing a bar partition in this set with a given $p$-bar-core and $q$-bar-core.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.231
Classification: 05E10, 05A17, 20C30
Keywords: representation theory, symmetric group, partitions, projective, bar-core
Yates, Dean 1

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