The Terwilliger Algebra of a Distance-Regular Graph that Supports a Spin Model

Let denote a distance-regular graph with vertex set X , diameter D ≥ 3, valency k ≥ 3, and assume supports a spin model W . Write W = ∑D i=0 ti Ai where Ai is the i th distance-matrix of . To avoid degenerate situations we assume is not a Hamming graph and ti ∈ {t0, −t0} for 1 ≤ i ≤ D. In an earlier paper Curtin and Nomura determined the intersection numbers of in terms of D and two complex parameters η and q . We extend their results as follows. Fix any vertex x ∈ X and let T = T (x) denote the corresponding Terwilliger algebra. Let U denote an irreducible T -module with endpoint r and diameter d . We obtain the intersection numbers ci (U ), bi (U ), ai (U ) as rational expressions involving r, d, D, η and q. We show that the isomorphism class of U as a T -module is determined by r and d. We present a recurrence that gives the multiplicities with which the irreducible T -modules appear in the standard module. We compute these multiplicites explicitly for the irreducible T -modules with endpoint at most 3. We prove that the parameter q is real and we show that if is not bipartite, then q > 0 and η is real.