Q-polynomial distance-regular graphs with a1=0 and a2≠0

AbstractLet Γ denote a Q-polynomial distance-regular graph with diameter D≥3 and intersection numbers a1=0, a2≠0. Let X denote the vertex set of Γ and let A∈MatX(C) denote the adjacency matrix of Γ. Fix x∈X and let A∗∈MatX(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of MatX(C) generated by A,A∗. We call T the Terwilliger algebra of Γ with respect to x. We show that up to isomorphism there exists a unique irreducible T-module W with endpoint 1. We show that W has dimension 2D−2. We display a basis for W which consists of eigenvectors for A∗. We display the action of A on this basis. We show that W appears in the standard module of Γ with multiplicity k−1, where k is the valency of Γ