Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra

Let Γ denote a distance-regular graph with diameter D≥3, intersection numbers ai, bi, ci and Bose–Mesner algebra M. For θ∈C∪∞ we define a one-dimensional subspace of M which we call M(θ). If θ∈C then M(θ) consists of those Y in M such that (A−θI)Y∈CAD, where A (resp. AD) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ=∞ then M(θ)=CAD. By a pseudo primitive idempotent for θ we mean a nonzero element of M(θ). We use these as follows. Let X denote the vertex set of Γ and fix x∈X. Let T denote the subalgebra of MatX(C) generated by A, E0∗,E1∗,…,ED∗, where Ei∗ denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the Terwilliger algebra. Let W denote an irreducible T-module. By the endpoint of W we mean min{i∣Ei∗W≠0}. W is called thin whenever dim(Ei∗W)≤1 for 0≤i≤D. Let V=CX denote the standard T-module. Fix 0≠v∈E1∗V with v orthogonal to the all ones vector. We define (M;v):={P∈M∣Pv∈ED∗V}. We show the following are equivalent: (i) dim(M;v)≥2; (ii) v is contained in a thin irreducible T-module with endpoint 1. Suppose (i), (ii) hold. We show (M;v) has a basis J, E where J has all entries 1 and E is defined as follows. Let W denote the T-module which satisfies (ii). Observe E1∗W is an eigenspace for E1∗AE1∗; let η denote the corresponding eigenvalue. Define η=−1−b1(1+η)−1 if η≠−1 and η=∞ if η=−1. Then E is a pseudo primitive idempotent for η.