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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:
 5e+30 Association schemes, strongly regular graphs
##### Keywords:
irreducible $$T$$-modules
Full Text:
##### References:
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