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The Terwilliger algebras of bipartite $$P$$- and $$Q$$-polynomial schemes. (English) Zbl 0924.05067
Let $$Y=(X,\{R_i\}_{0\leq i\leq D})$$ be a bipartite $$P$$- and $$Q$$-polynomial association scheme. Then $$A_0,\dots,A_D$$ form a basis for the Bose-Mesner algebra $$M$$ of $$Y$$ ($$A_i$$ is the $$i$$th associate matrix of $$Y$$). We now recall the dual Bose-Mesner algebra $$M^*$$ of $$Y$$. Fix any $$x\in X$$. For integer $$i$$ ($$0\leq i\leq D$$) let $$E_i^*=E_i^*(x)$$ denote the diagonal matrix with the rows and columns indexed by $$X$$ and $$(E_i^*)yy=1$$, if $$xy\in R_i$$, $$(E_i^*)yy=0$$, if $$xy\notin R_i$$. We refer to $$E_i^*$$ as the $$i$$th dual idempotent of $$Y$$ with respect to $$x$$. It follows that the matrices $$E_0^*,\dots,E_D^*$$ form a basis for a algebra $$M^*=M^*(x)$$ known as the dual Bose-Mesner algebra of $$Y$$ with respect to $$x$$. The subalgebra $$T=T(x)$$ of $$\text{Mat}_X({\mathbb{C}})$$ generated by $$M$$ and $$M^*$$ is called the Terwilliger algebra of $$Y$$ with respect to $$x$$. Let $$V$$ denote the vector space $${\mathbb{C}}^X$$ (column vectors). Then $$\text{Mat}_X({\mathbb{C}})$$ acts on $${\mathbb{C}}^X$$ by left multiplication. By a $$T$$-module we mean a subspace $$W$$ of $$V$$ such that $$TW\subseteq W$$. The author proves that any irreducible $$T$$-module $$W$$ is both thin and dual thin in the sense of Terwilliger. The isomorphism class of $$W$$ as a $$T$$-module is determined by two parameters, the endpoint and diameter of $$W$$. A recurrence which gives the multiplicities with which the irreducible $$T$$-modules occur in standard module $$V$$ is found.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
##### Keywords:
association schemes; Terwilliger algebra
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##### References:
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