-polynomial distance-regular graphs with and

Let Γ denote a Q -polynomial distance-regular graph with diameter D ≥ 3 and intersection numbers a 1 = 0 , a 2 ≠ 0 . Let X denote the vertex set of Γ and let A ∈ Mat X ( C ) denote the adjacency matrix of Γ . Fix x ∈ X and let A ∗ ∈ Mat X ( C ) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of Mat X ( 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 2 D − 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 .