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

Let Γ denote a distance-regular graph with vertex set X, diameter D ≥ 3, valency k ≥ 3, and assume Γ supports a spin model W. Write W = ∑i = 0DtiAi where Ai is the ith distance-matrix of Γ. To avoid degenerate situations we assume Γ 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 Γ 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 Γ is not bipartite, then q > 0 and η is real.