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 = _{i = 0}^D t_i A_i where A_i is the ith distance-matrix of . To avoid degenerate situations we assume is not a Hamming graph and t_i {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 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 c_i(U), b_i(U), a_i(U) as rational expressions involving r, d, D, 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 is real.AMS 2000 Subject Classification: Primary 05E30