緊密距離正則圖之研究

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Abstract: Let Γ = (X, R) denote a distance-regular graph with diameter D ≥ 3. Let M denote the Bose-Mesner algebra of Γ. Fix a base vertex x ∈ X, and let T denote the Ter-williger algebra of Γ with respect to x. Let E∗ i

(0 ≤ i ≤ D) denote the ith dual idempotent of Γ with respect to x. Let V denote the stan-dard T-module of Γ. Suppose v ∈ E∗

1V is a

nonzero vector which is orthogonal to the ir-reducible T-module of endpoint 0. Let (M; v) denote the subspace {P ∈ M|P v ∈ E∗

DV } of

M. Then the following (i)-(ii) are equivalent.

(i) (M; v) has dimension 2.

(ii) Mv is a thin irreducible T-module of endpoint 1.

Furthermore, suppose (i)-(ii) hold, and let α be the eigenvalue of E∗

1AE1∗ associated with

with v. Then (M; v) has a basis {J, E}, where

E is a pseudo idempotent associated with

e α = ( ∞ α = −1, −1 − b1 1+α otherwise.

Keywords: Terwilliger algebra; dual idem-potent; pseudo idempotent.

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