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The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two. (English) Zbl 1136.05076

Let \(\Gamma\) be a distance-regular graph with diameter \(D\), distance matrices \(A_i\) and eigenvalues \(\theta_0>\theta_1>\dots >\theta_D\). Fix a base vertex \(x\) of \(\Gamma\). Let \(T=T(x)\) denote the subalgebra of \(Mat_X({\mathbb C})\) generated by \(A,E_0^*,\dots ,E_D^*\), where \(A=A_1\) and \(E_i^*\) denotes the projection onto the \(i\)th subconstituent of \(\Gamma\) with respect to \(x\). \(T\) is called the Terwilliger algebra of \(\Gamma\) with respect to \(x\). There exist the unique irreducible \(T\)-module \(V_0\) with endpoint 0 (\(V_1\) with endpoint 1). Both \(V_0\) and \(V_1\) are thin. Assume \(\Gamma\) is bipartite and \(W\) is thin irreducible \(T\)-module with endpoint 2. Then \(E_2^*W\) is a one-dimensional eigenspace for \(E_2^*A_2E_2^*\) with eigenvalue \(\eta\) and \(\tilde \theta_1\leq \eta\leq \tilde \theta_d\), where \(\tilde \theta_i=-1-b_2b_3(\theta_i^2-b_2)^{-1}\) and \(d=[D/2]\). To describe the structure of \(W\) we distinguish four cases:
(i)
\(\eta=\tilde \theta_1\),
(ii)
\(D\) is odd and \(\eta=\tilde \theta_d\),
(iii)
\(D\) is even and \(\eta=\tilde \theta_d\),
(iv)
\(\tilde \theta_1<\eta<\tilde \theta_d\).
Cases \((i)\) and \((ii)\) were investigated earlier. It is showed that \({\text{ dim}}W=D-1-e\) (\(e=1\) in the case \((iii)\), \(e=0\) in the case \((iv)\)), \(W\) has orthogonal basis \(E_iv (i\in S)\), where \(v\) – a nonzero vector of \(E_2^*W\), \(E_i\) – primitive idempotent \(A\), associated with \(\theta_i\), and \(S=\{1,2,\dots D-1\}-\{d\}\) in the case \((iii)\), \(S=\{1,2,\dots D-1\}\) in the case \((iv)\), moreover \(W\) has orthogonal basis \(E_{i+2}^*A_iv (0\leq i\leq D-2-e)\). It is founded square-norm of each vector of two bases and transition matrix relating this two bases.

MSC:

05E30 Association schemes, strongly regular graphs
20D15 Finite nilpotent groups, \(p\)-groups
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