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The Terwilliger algebra of a distance-regular graph that supports a spin model. (English) Zbl 1064.05152

Summary: Let \(\Gamma\) denote a distance-regular graph with vertex set \(X\), diameter \(D \geq 3\), valency \(k \geq 3\), and assume \(\Gamma\) supports a spin model \(W\). Write \(W = \sum_{i=0}^{D} t_{i}A_{i}\) where \(A_{i}\) is the \(i\)th distance-matrix of \(\Gamma\). To avoid degenerate situations we assume \(\Gamma\) is not a Hamming graph and \(t_{i} \notin \{t_{0}, -t_{0}\}\) for \(1 \leq i \leq D\). In an earlier paper Curtin and Nomura determined the intersection numbers of \(\Gamma\) in terms of \(D\) and two complex parameters \(\eta\) and \(q\). We extend their results as follows. Fix any vertex \(x \in X\) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Let \(U\) denote an irreducible \(T\)-module with endpoint \(r\) and diameter \(d\). We obtain the intersection numbers \(c_i(U)\), \(b_i(U)\), \(a_i(U)\) as rational expressions involving \(r,d,D,\eta\) and \(q\). We show that the isomorphism class of \(U\) as a \(T\)-module is determined by \(r\) and \(d\). We present a recurrence that gives the multiplicities with which the irreducible \(T\)-modules appear in the standard module. We compute these multiplicites explicitly for the irreducible \(T\)-modules with endpoint at most 3. We prove that the parameter \(q\) is real and we show that if \(\Gamma\) is not bipartite, then \(q > 0\) and \(\eta\) is real.

MSC:

05E30 Association schemes, strongly regular graphs
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