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 and denote Bose–Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to . Also suppose that is imprimitive with respect to a subset of Hadamard idempotents, so is dual imprimitive with respect to the subset of primitive idempotents, where is the formal duality associated with ψ. Let denote the block Bose–Mesner algebra of on the block containing p, and let denote the quotient Bose–Mesner algebra of with respect to . Then there is a hyper-duality from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to .     

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V satisfying both conditions below:

1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

2. [(ii)] There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.

We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A^* on V, there exists a sequence of scalars β,γ,γ^*,,^* taken from such that both

where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.     

A class of amply regular graphs related to the subconstituents of a dual polar graph

The connected components of the induced graphs on each subconstituent of the dual polar graph of the odd dimensional orthogonal spaces over a finite field are shown to be amply regular. The connected components of the graphs on the second and third subconstituents are shown to be distance-regular by elementary methods.     

The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph

Let denote a Q-polynomial distance-regular graph with diameter at least three and standard module V. We introduce two direct sum decompositions of V. We call these the displacement decomposition for and the split decomposition for . We describe how these decompositions are related.