Inheritance of hyper-duality in imprimitive Bose-Mesner algebras

We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose-Mesner algebras associated with a hyper-dual pair of imprimitive Bose-Mesner algebras. Let M and M̃ denote Bose-Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of M with respect to p to the subconstituent algebra of M̃ with respect to p̃. Also suppose thatM is imprimitive with respect to a subset I of Hadamard idempotents, so M̃ is dual imprimitive with respect to the subset Ψ(I) of primitive idempotents, where Ψ : M → M̃ is the formal duality associated with ψ. Let B denote the block Bose-Mesner algebra ofM on the block containing p, and let Q̃ denote the quotient Bose-Mesner algebra of M̃ with respect to Ψ(I). Then there is a hyper-duality from the subconstituent algebra of B with respect to p to the subconstituent algebra of Q̃ with respect to p̃.