A generalisation of core partitions

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 λ is at most the s-weight of λ. Then we consider the set 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 A ˜ s − 1 × 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.