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

Let Γ be a Q-polynomial distance-regular graph with vertex set X, diameter D⩾3 and adjacency matrix A. Fix x∈X and let A∗=A∗(x) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T=T(x) is the subalgebra of MatX(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 Γ.Our results are summarized as follows. Let be a Q-polynomial ordering of the primitive idempotents of Γ and let be the dual primitive idempotents of Γ with respect to x. Let r,t and d be the endpoint, dual endpoint and diameter of W, respectively. Let be nonzero vectors in EtW and respectively. We show that and 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 . We show that for 0⩽i⩽d, and Next, we show that and 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. We call these the intersection numbers and dual intersection numbers of W. Using these numbers, we compute all inner products involving the standard and dual standard bases. We also use these numbers to define two normalizations ui,vi (resp. ) for pi (resp. ). Using the orthogonality of the standard and dual standard bases, we show that for each of the sequences the polynomials involved are orthogonal and we display the orthogonality relations. We also show that each of the sequences satisfy a three-term recurrence and a relation known as the Askey–Wilson duality. We then turn our attention to two more bases for W. We find the matrix representations of A and A∗ with respect to these bases. From the entries of these matrices we obtain two sequences of scalars known as the first split sequence and second split sequence of W. We associate with W a sequence of scalars called the parameter array. This sequence consists of the eigenvalues of the restriction of A to W, the eigenvalues of the restriction of A∗ to W, the first split sequence of W and the second split sequence of W. We express all the scalars and polynomials associated with W in terms of its parameter array. We show that the parameter array of W is determined by r,t,d and one more free parameter. From this we conclude that the isomorphism class of W is determined by these four parameters. Finally, we apply our results to the case in which Γ has q-Racah type or classical parameters.