The Terwilliger algebra of a distance-regular graph of negative type

Let Gamma denote a distance-regular graph with diamter D >= 3. Assume Gamma has classical parameters (D, b, alpha, beta) with b < -1. Let X denote the vertex set of Gamma and let A is an element of Mat(X)(C) denote the adjacency matrix of Gamma. Fix x is an element of X and let A* is an element of Mat(X)(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of Mat(X)(C) generated by A, A*. We call T the Terwilliger albgebra of Gamma with respect to x. We show that up to isomorphism there exist exactly two irreducible T-modules with endpoint 1; their dimensions and D and 2D - 2. For these T-modules we display a basis consisting of eigenvectors for A*, and for each basis we give the action of A. (C) 2008 Elseiver Inc. All rights reserved.