The subconstituent algebra of a distance-regular graph; thin modules with endpoint one
The author considers a distance-regular graph \(\Gamma\) with diameter \(D\) and eigenvalues \(\theta_0>\dots >\theta_D\). Let \(x\) be a vertex of \(\Gamma\). Let \(T=T(x)\) denote the subalgebra of \(\text{Mat}_X({\mathbb{C}})\) generated by \(A,E_0^*,...,E_D^*\), where \(A\) is the adjacency matrix of \(\Gamma\) and \(E_i^*\) is the projection onto the \(i\)th subconstituent 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\(\{i\;|\;E_i^*W\neq 0\}\). The author describes the thin irreducible \(T\)-modules with endpoint 1. Let \(W\) be a thin irreducible \(T\)-module with endpoint 1. Then \(E_1^*W\) is a one-dimensional eigenspace for \(E_1^*AE_1^*\). Let \(v\) be a nonzero vector in \(E_1^*W\), \(\eta\) denote the corresponding eigenvalue and \(\tilde \theta_i=-1-b_1(1+\theta_i)^{-1}\). Then \(\tilde \theta_1\leq \eta\leq \tilde \theta_D\). The extremal cases \(\eta= \tilde \theta_1\) and \(\eta=\tilde \theta_D\) were consider by Terwilliger and Go (preprint), the results are summarized in \S\ 9. For \(\tilde \theta_1<\eta<\tilde \theta_D\) the module \(W\) has two orthogonal bases \(E_iv\) \((1\leq i\leq D)\), where \(E_i\) denotes the primitive idempotent of \(A\) associated with \(\theta_i\), and \(E_{i+1}^*A_iv\) \((0\leq i\leq D-1)\) (Theorems 11.1 and 11.4). The transition matrix relating this two bases for \(W\) is calculated.
scientific article; zbMATH DE number 1864760