Crested Products of Association Schemes

In this paper, we define 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 Qr for r = 1, 2, each having an inherent partition Fr (that is, a partition whose equivalence relation is a union of adjacency relations in the association scheme), we define a product of the two schemes, which reduces to the direct product if F1 = U1 or F2 = E2, and to the wreath product if F1 = E1 and F2 = U2, where Er and Ur are the relation of equality and the universal relation on Qr. We calculate the character table of the crested product, and show that if the two schemes Q1 and Q2 have formal duals, then so does their crested product (and we give a simple description of this dual). We make an analogous definition for permutation groups with intransitive normal subgroups, and show 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 works as expected only if the inherent partitions (or orbit partitions) form a distributive lattice. We conclude with some open problems.