A generalisation of core partitions

TL;DR

This paper generalizes Olsson's theorem on the relationship between s-cores and t-cores of partitions, introduces the concept of [s:t]-cores, and explores their structure and symmetries under Coxeter group actions.

Contribution

It extends Olsson's theorem to show the s-weight of t-cores is bounded by that of the original partition and introduces [s:t]-cores with a rich algebraic structure.

Findings

The s-weight of t-cores is at most that of the original partition.

[s:t]-cores form a union of finitely many Coxeter group orbits.

The structure of [s:t]-cores suggests directions for future research.

Abstract

Suppose $s$ and $t$ are coprime natural numbers. A theorem of Olsson says that the $t$-core of an $s$-core partition is again an $s$-core. We generalise this theorem, showing that the $s$-weight of the $t$-core of a partition $\lambda$ is at most the $s$-weight of $\lambda$. Then we consider the set $\mathcal C_{s:t}$ of partitions for which equality holds, which we call $[s{:}t]$-cores; this set has interesting structure, and we expect that it will be the subject of future study. We show that the set of $[s{:}t]$-cores is a union of finitely many orbits for an action of a Coxeter group of type $\tilde A_{s-1}\times\tilde A_{t-1}$ on the set of partitions. We also consider the problem of constructing an $[s{:}t]$-core with specified $s$-core and $t$-core.