The Terwilliger algebras of bipartite P- and Q-polynomial schemes

Abstract

Let Y denote a D-class symmetric association scheme with D ⩾ 3, and suppose Y is bipartite P- and Q-polynomial. Let T denote the Terwilliger algebra with respect to any vertex x. 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 endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the irreducible T-modules occur in the standard module. Using this recurrence, we produce formulas for the multiplicities of the irreducible T-modules with endpoint at most four.