Tight distance-regular graphs and the subconstituent algebra

Abstract

We consider a distance-regular graph Γ with diameter D ≥ 3, intersection numbers ai, bi, ci and eigenvalues k = θ0 > θ1 > ⋯ > θD. Let X denote the vertex set of Γ and fix x ∈ X. Let T = T (x) denote the subalgebra of MatX (ℂ) generated by A, E0*, E1*,...,ED*, where A denotes the adjacency matrix of Γ and Ei* denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim E i* W ≤ 1 for 0 ≤ i ≤ D. By the endpoint of W we mean min{i|Ei* W ≠ 0}. Let W denote a thin irreducible T-module with endpoint 1. Observe E1* W is a one-dimensional eigenspace for E1* A E1*; let η denote the corresponding eigenvalue. We call η the local eigenvalue of W. It is known θ̃1 ≤ η ≤ θ̃D where θ⇔1 = -1 - b1 (1 + θ1) -1 and θ̃D = -1 - b1(1 + θD)-1. Let n = 1 or n = D and assume η = θ̃n. We show the dimension of W is D - 1. Let υ denote a nonzero vector in E1* W. We show W has a basis E i υ (1 ≤ i ≤ D, i ≠ n), where Ei denotes the primitive idempotent of A associated with θi. We show this basis is orthogonal (with respect to the Hermitean dot product) and we compute the square norm of each basis vector. We show W has a basis E i+1*Ai υ (0 ≤ i ≤ D - 2), where Ai denotes the ith distance matrix for Γ. We find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square norm of each basis vector. We find the transition matrix relating our two bases for W. For notational convenience, we say Γ is 1-thin with respect to x whenever every irreducible T-module with endpoint 1 is thin. Similarly, we say Γ is tight with respect to x whenever every irreducible T-module with endpoint 1 is thin with local eigenvalue θ̃1 or θ̃D. In [J. Algebr. Comb., 12, (2000), 163-197] Jurišić, Koolen and Terwilliger showed (↘1 + k/a1 + 1)(θD + k/a1 + 1) ≥ ka 1b1/a1 + 1)2. They defined Γ to be tight whenever Γ is nonbipartite and equality holds above. We show the following are equivalent: (i) Γ is tight; (ii) γ is tight with respect to each vertex; (iii) Γ is tight with respect to at least one vertex. We show the following ate equivalent: (i) Γ is tight; (ii) Γ is nonbipartite, aD = 0, and Γ is 1-thin with respect to each vertex; (iii) Γ is nonbipartite, aD = 0, and Γ is 1-thin with respect to at least one vertex. © 2002 Elsevier Science Ltd. All rights reserved.