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

Paul Terwilliger*, Chih-wen Weng

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


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∈ C AD, where A (resp. AD) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ=∞ then M (θ)=C AD. 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*, E 1*, ...,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 t he jstandard 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 η̃=-1-b1(1+η)-1 if η≠-1 and η̃=∞ if η=-1. Then E is a pseudo primitive idempotent for η̃.

Original languageEnglish
Pages (from-to)287-298
Number of pages12
JournalEuropean Journal of Combinatorics
Issue number2
StatePublished - 1 Feb 2004


  • Distance-regular graph
  • Pseudo primitive idempotent
  • Subconstituent algebra
  • Terwilliger algebra


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