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

We consider a bipartite distance-regular graph image with diameter image, valency image, intersection numbers image, distance matrices image, and eigenvalues image. Let X denote the vertex set of image and fix image. Let image denote the subalgebra of image generated by image, where image and image denotes the projection onto the imageth subconstituent of image with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of image with respect to x. An irreducible T-module W is said to be thin whenever image for image. By the endpoint of W we mean image. Assume W is thin with endpoint 2. Observe image is a one-dimensional eigenspace for image; let image denote the corresponding eigenvalue. It is known image where image, and image. To describe the structure of W we distinguish four cases: (i) image; (ii) D is odd and image; (iii) D is even and image; (iv) image. We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694–1721]. Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W is image where image in case (iii) and image in case (iv). Let image denote a nonzero vector in image. We show W has a basis image, where image denotes the primitive idempotent of A associated with image and where the set S is image in case (iii) and image in case (iv). 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 image, and 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.