The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme Original Research Article

Let image denote a image-class symmetric association scheme with image, and suppose image is almost-bipartite P- and Q-polynomial. Let image denote a vertex of image and let image denote the corresponding Terwilliger algebra. We prove that any irreducible image-module image is both thin and dual thin in the sense of Terwilliger. We produce two bases for image and describe the action of image on these bases. We prove that the isomorphism class of image as a image-module is determined by two parameters, the dual endpoint and diameter of image. We find a recurrence which gives the multiplicities with which the irreducible image-modules occur in the standard module. We compute this multiplicity for those irreducible image-modules which have diameter at least image.