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Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra. (English) Zbl 1035.05104
A distance-regular graph $$\Gamma$$ with diameter $$D$$ and Bose-Mesner algebra $${\mathbb M}$$ is considered. Let $$A_i$$ be the $$i$$-adjacency matrix of $$\Gamma$$. For $$\theta\in {\mathbb C}$$ let $${\mathbb M}(\theta)$$ consist of those $$Y\in {\mathbb M}$$ such that $$(A-\theta I)Y\in {\mathbb C}A_D$$ and $${\mathbb M}(\infty)={\mathbb C}A_D$$. By a pseudo primitive idempotent for $$\theta\in {\mathbb C}\cup \{\infty\}$$ we mean a nonzero element of $${\mathbb M}(\theta)$$. Let $$X$$ denote the vertex set of $$\Gamma$$ and $$x\in X$$. Let $$T=T(x)$$ denote the subalgebra of $$\text{Mat}_X({\mathbb C})$$ generated by $$A,E_0^*,\dots,E_D^*$$, where $$A$$ is the adjacency matrix of $$\Gamma$$ and $$E_i^*$$ is the projection onto the $$i$$th subconstituent of $$\Gamma$$ with respect to $$x$$. $$T$$ is called the Terwilliger algebra of $$\Gamma$$ with respect to $$x$$. An irreducible $$T$$-module $$W$$ is thin whenever dim $$E_i^*W\leq 1$$ for $$0\leq i\leq D$$. The endpoint of $$W$$ is min$$\{i\;| \;E_i^*W\neq 0\}$$. Let $$V={\mathbb C}^X$$ denote the standard $$T$$-module. Fix $$0\neq v\in E_1^*V\cap {\mathbf 1}^\bot$$ and define $$({\mathbb M},v)=\{P\in {\mathbb M}\;| \;Pv\in E_D^*V\}$$. The following are equivalent: (i) dim($${\mathbb M},v)\geq 2$$; (ii) $$v$$ is contained in a thin irreducible $$T$$-module with endpoint 1 (Theorem 1.1). Let $$W$$ be the $$T$$-module which satisfies (ii). Then $$E_1^*W$$ is a one-dimensional eigenspace for $$E_1^*AE_1^*$$. Let $$\eta$$ denote the corresponding eigenvalue and $$\tilde \eta=-1-b_1(1+\eta)^{-1}$$ if $$\eta \neq -1$$ and $$\tilde \eta=\infty$$ if $$\eta=-1$$. Then $$({\mathbb M},v)$$ has a basis $$(J,E)$$, where $$J$$ has all entries 1 and $$E$$ is a pseudo primitive idempotent for $$\tilde \eta$$ (Theorem 1.2).

##### MSC:
 5e+30 Association schemes, strongly regular graphs
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