CRESTED PRODUCTS OF ASSOCIATION SCHEMES

Abstract

The paper defines a new type of product of association schemes (and of the related objects, permutation groups and orthogonal block structures), which generalizes the direct and wreath products (which are referred to as ‘crossing’ and ‘nesting’ in the statistical literature). Given two association schemes $\mathcal{Q}_r$ for $r\,{=}\,1,2$, each having an inherent partition $F_r$ (that is, a partition whose equivalence relation is a union of adjacency relations in the association scheme), a product of the two schemes is defined, which reduces to the direct product if $F_1=U_1$ or $F_2=E_2$, and to the wreath product if $F_1 = E_1$ and $F_2 = U_2$, where $E_r$ and $U_r$ are the relation of equality and the universal relation on $\mathcal{Q}_r$. The character table of the crested product is calculated, and it is shown that, if the two schemes $\mathcal{Q}_1$ and $\mathcal{Q}_2$ have formal duals, then so does their crested product (and a simple description of this dual is given). An analogous definition for permutation groups with intransitive normal subgroups is created, and it is shown that the constructions for association schemes and permutation groups are related in a natural way.

The definition can be generalized to association schemes with families of inherent partitions, or permutation groups with families of intransitive normal subgroups. This time the correspondence is not so straightforward, and it works as expected only if the inherent partitions (or orbit partitions) form a distributive lattice.

The paper concludes with some open problems.