The t-core of an s-core

TL;DR

This paper explores the properties of the t-core of s-core partitions, demonstrating its preservation under certain group actions and providing new proofs for existing results about core partitions.

Contribution

It offers new proofs of Olsson's and Vandehey's results, and analyzes affine symmetric group actions that preserve the t-core of s-core partitions.

Findings

The t-core of an s-core is itself an s-core when s and t are coprime.

Certain affine symmetric group actions preserve the t-core of s-core partitions.

Existence of a simultaneous s- and t-core containing all others.

Abstract

We consider the $t$-core of an $s$-core partition, when $s$ and $t$ are coprime positive integers. Olsson has shown that the $t$-core of an $s$-core is again an $s$-core, and we examine certain actions of the affine symmetric group on $s$-cores which preserve the $t$-core of an $s$-core. Along the way, we give a new proof of Olsson's result. We also give a new proof of a result of Vandehey, showing that there is a simultaneous $s$- and $t$-core which contains all others.