Rook Theory andt-Cores

Iftis a positive integer, then a partition of a non-negative integernis at-core if none of the hook numbers of the associated Ferrers–Young diagram is a multiple oft. These partitions arise in the representation theory of finite groups and also in the theory of class numbers. We prove that ift=2, 3, or 4, then two differentt-cores are rook equivalent if and only if they are conjugates. In the special case whent=4, sincec4(n)=12h(−32n−20) when 8n+5 is square-free, the above result suggests a new method of approaching Gaussʹ class number problem for these discriminants. Unlike the cases where 2⩽t⩽4, it turns out that whent⩾5 there are distinct rook equivalentt-cores which are not conjugates. In fact, we conjecture that for all sucht, there exists a constantN(t) for which every integern⩾N(t) has the property that there exists a pair of distinct rook equivalentt-cores ofnwhich are not conjugates.