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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).

05E30 Association schemes, strongly regular graphs
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