The Structure of Nonthin Irreducible T-modules of Endpoint 1: Ladder Bases and Classical Parameters

Building on the work of Terwilliger, we find the structure of nonthin irreducible T-modules of endpoint 1 for P- and Q-polynomial association schemes with classical parameters. The isomorphism class of such a given module is determined by the intersection numbers of the scheme and one additional parameter which must be an eigenvalue for the first subconstituent graph. We show that these modules always have what we call a ladder basis, and find the structure explicitly for the bilinear, Hermitean, and alternating forms schemes.