Associative algebras, Combinatorial analysis, Irreducible polynomials
Let Y denote a D-class symmetric association scheme with D≥3, and suppose Y is almost-bipartite P- and Q-polynomial. Let x denote a vertex of Y and let T=T(x) denote the corresponding Terwilliger algebra. We prove that any irreducible T-module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the dual endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the irreducible T-modules occur in the standard module. We compute this multiplicity for those irreducible T-modules which have diameter at least D−3.
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Published as: John S. Caughman, Mark S. MacLean, Paul M. Terwilliger (2005). The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme, Discrete Mathematics, Volume 292, Issues 1–3, Pages 17-44.