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

Abstract

Let Γ denote a distance-regular graph with diameter D≥3, intersection numbers a i , b i , c i 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∈ CA D , where A (resp. A D ) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ=∞ then M (θ)= CA D . 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 Mat X( 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≠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≠v∈E 1 ∗V with v orthogonal to the all ones vector. We define ( M ;v):={P∈ M ∣Pv∈E D ∗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 E 1 ∗W is an eigenspace for E 1 ∗AE 1 ∗ ; let η denote the corresponding eigenvalue. Define η=−1−b 1(1+η) −1 if η≠−1 and η=∞ if η=−1. Then E is a pseudo primitive idempotent for η .