Inheritance of hyper-duality in imprimitive Bose–Mesner algebras

Abstract

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 M ˜ 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 M ˜ with respect to p ˜ . Also suppose that M is imprimitive with respect to a subset I of Hadamard idempotents, so M ˜ is dual imprimitive with respect to the subset Ψ ( I ) of primitive idempotents, where Ψ : M → M ˜ is the formal duality associated with ψ . Let B denote the block Bose–Mesner algebra of M on the block containing p, and let Q ˜ denote the quotient Bose–Mesner algebra of M ˜ 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 Q ˜ with respect to p ˜ .