The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme

Details The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme

2005-08-22

arxiv.org/abs/math/0508401v1

Mathematics

Combinatorics

22 pages

Scientific paper

Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, 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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