A further generalisation of bar-core partitions

TL;DR

This paper generalizes Olsson's theorem on bar-cores, showing the relationship between p- and q-bar-cores and exploring the structure of partitions where equality holds, including an algorithm for construction.

Contribution

It extends Olsson's result to a broader setting, analyzing the structure of bar partitions and their cores with a new algebraic and combinatorial approach.

Findings

The p-bar-weight of the q-bar-core is at most that of the original partition.

The set of partitions with equal p- and q-bar-weights forms a union of Coxeter group orbits.

An algorithm is provided to construct partitions with specified p- and q-bar-cores.

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 $\lambda$ 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 $\tilde C_{\tfrac{(p-1)}{2}}\times\tilde C_{\tfrac{(q-1)}{2}}$. We also provide an algorithm for constucting a bar partition in this set with a given $p$-bar-core and $q$-bar-core.