Taut distance-regular graphs and the subconstituent algebra

We consider a bipartite distance-regular graph Γ with diameter D ≥ 4 and valency k ≥ 3. Let X denote the vertex set of Γ and fix x ∈ X. Let Γ22 denote the graph with vertex set X̆ = {y ∈ X | ∂(x, y) = 2}, and edge set R̆ = {yz | y, z ∈ X̆, ∂(y, z) = 2}, where ∂ is the path-length distance function for Γ. The graph Γ22 has exactly k2 vertices, where k2 is the second valency of Γ. Let η1, η2, . . . , ηk2 denote the eigenvalues of the adjacency matrix of Γ22; we call these the local eigenvalues of Γ. Let A denote the adjacency matrix of Γ. We obtain upper and lower bounds for the local eigenvalues in terms of the intersection numbers of Γ and the eigenvalues of A. Let T = T (x) denote the subalgebra of MatX(C) generated by A,E ∗ 0 , E ∗ 1 , . . . , E ∗ D, where for 0 ≤ i ≤ D, E ∗ i represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T -module W is said to be thin whenever dimE∗ i W ≤ 1 for 0 ≤ i ≤ D. By the endpoint of W we mean min{i|E∗ i W 6= 0}. We give a detailed description of the thin irreducible T -modules that have endpoint 2 and dimension D − 3. In [Discrete Math., 225(2000), 193–216] MacLean defined what it means for Γ to be taut. We obtain three characterizations of the taut condition, each of which involves the local eigenvalues or the above T -modules.