Structure of thin irreducible modules of a Q-polynomial distance-regular graph

Let Gamma be a Q-polynomial distance-regular graph with vertex set X, diameter D >= 3 and adjacency matrix A. Fixx is an element of X and let A* = A* (x) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T = T(x) is the subalgebra of Mat(x)(C) generated by A and A*. Let W denote a thin irreducible T-module. It is known that the action of A and A* on W induces a linear algebraic object known as a Leonard pair. Over the past decade, many results have been obtained concerning Leonard pairs. In this paper, we apply these results to obtain a detailed description of W. In our description, we do not assume that the reader is familiar with Leonard pairs. Everything will be proved from the point of view of Gamma. Our results are summarized as follows. Let {E(i)}(i=0)(D) be a Q-polynomial ordering of the primitive idempotents of Gamma and let {E(i)(*)}(i=0)(D) be the dual primitive idempotents of Gamma with respect to x. Let r, t and d be the endpoint, dual endpoint and diameter of W, respectively. Let u and v be nonzero vectors in E(t)W and E(r)(*)W, respectively. We show that {E(r+i)(*)A(i)v}(i=0)(d) and {E(t+i)A(*i)u}(i=0)(d) are bases for W that are orthogonal with respect to the standard Hermitian dot product. We display the matrix representations of A and A* with respect to these bases. We associate with W two sequences of polynomials {pi}(i=0)(d) and {p(i)*}(i=0)(d) We show that for 0 <= i <= d, p(i)(A)v = E(r+i)(*)A(i)v and p(i)* (A*)u = E(t+i)A(*i) u. Next, we show that {E(r+i)*u}(i=0)(d) and {E(t+i)v}(i=0)(d) are orthogonal bases for W, we call these the standard basis and dual standard basis for W, respectively. We display the matrix representations of A and A* with respect to these bases. The entries in these matrices will play an important role in our theory.