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

Original language | English |
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Pages (from-to) | 287-298 |

Number of pages | 12 |

Journal | European Journal of Combinatorics |

Volume | 25 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2004 |

## Keywords

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