A generalisation of core partitions

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.