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 MM and M˜ denote Bose–Mesner algebras. Suppose there is a hyper-duality ψψ from the subconstituent algebra of MM with respect to p   to the subconstituent algebra of M˜ with respect to p˜. Also suppose that MM is imprimitive with respect to a subset II of Hadamard idempotents, so M˜ is dual imprimitive with respect to the subset Ψ(I)Ψ(I) of primitive idempotents, where Ψ:M→M˜ is the formal duality associated with ψψ. Let BB denote the block Bose–Mesner algebra of MM on the block containing p  , and let Q˜ denote the quotient Bose–Mesner algebra of M˜ with respect to Ψ(I)Ψ(I). Then there is a hyper-duality from the subconstituent algebra of BB with respect to p   to the subconstituent algebra of Q˜ with respect to p˜.