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 1 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, E ∗ 0 , E ∗ 1 , · · · , E ∗ D, where E ∗ i 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|E i W 6= 0}. W is called thin whenever dim(E i W ) ≤ 1 for 0 ≤ i ≤ D. Let V = C X denote the standard T-module. Fix 0 6= v ∈ E 1V with v orthogonal to the all 1’s vector. We define (M; v) := {P ∈ M|Pv ∈ E DV }. 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 E 1W is an eigenspace for E ∗ 1AE ∗ 1 ; let η denote the corresponding eigenvalue. Define η̃ = −1− b1(1 + η) −1 if η 6= −1 and η̃ = ∞ if η = −1. Then E is a pseudo primitive idempotent for η̃.