The subconstituent algebra of a distance-regular graph; thin modules with endpoint one

Abstract

We consider a distance-regular graph Γ with diameter D⩾3, intersection numbers a i , b i , c i and eigenvalues θ 0> θ 1>⋯> θ D . Let X denote the vertex set of Γ and fix x∈ X. Let T= T( x) denote the subalgebra of Mat X( C) generated by A, E * 0, E * 1,…, E * D , where A denotes the adjacency matrix of Γ and E * i 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 * iW⩽1 for 0⩽ i⩽ D. By the endpoint of W we mean min{i | E * iW≠0} . We describe the thin irreducible T-modules with endpoint 1. Let W denote a thin irreducible T-module with endpoint 1. Observe E * 1 W is a one-dimensional eigenspace for E * 1 AE * 1; let η denote the corresponding eigenvalue. It is known θ ̃ 1⩽η⩽ θ ̃ D where θ ̃ 1=−1−b 1(1+θ 1) −1 and θ ̃ D=−1−b 1(1+θ D) −1 . For η= θ ̃ 1 and η= θ ̃ D the structure of W was worked out by Go and the present author [Tight distance-regular graphs and the subconstituent algebra, preprint]. For θ ̃ 1<η< θ ̃ D we obtain the following results. We show the dimension of W is D. Let v denote a nonzero vector in E * 1 W. We show W has a basis E i v (1⩽ i⩽ D), where E i denotes the primitive idempotent of A associated with θ i . We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square norm of each basis vector. We show W has a basis E * i+1 A i v (0⩽ i⩽ D−1), where A i 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.