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

Abstract

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 = 0D ti Ai where Ai is the ith 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 ? 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, ? 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.