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

We consider a distance-regular graph Γ with diameter Dgreater-or-equal, slanted3, intersection numbers ai,bi,ci and eigenvalues θ0>θ1>cdots, three dots, centered>θD. Let X denote the vertex set of Γ and fix xset membership, variantX. Let T=T(x) denote the subalgebra of image 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*iWless-than-or-equals, slant1 for 0less-than-or-equals, slantiless-than-or-equals, slantD. 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*1W is a one-dimensional eigenspace for E*1AE*1; let η denote the corresponding eigenvalue. It is known image where image and image . For image and image the structure of W was worked out by Go and the present author [Tight distance-regular graphs and the subconstituent algebra, preprint]. For image we obtain the following results. We show the dimension of W is D. Let v denote a nonzero vector in E*1W. We show W has a basis Eiv (1less-than-or-equals, slantiless-than-or-equals, slantD), where Ei denotes the primitive idempotent of A associated with θi. 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+1Aiv (0less-than-or-equals, slantiless-than-or-equals, slantD−1), 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.