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

Let Y denote a D-class symmetric association scheme with D ≥ 3, and suppose Y is almostbipartite Pand Q-polynomial. Let x denote a vertex of Y and let T = T (x) denote the corresponding Terwilliger algebra. We prove that any irreducible T -module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T -module is determined by two parameters, the dual endpoint and diameter of W . We find a recurrence which gives the multiplicities with which the irreducible T -modules occur in the standard module. We compute this multiplicity for those irreducible T -modules which have diameter at least D − 3.