## 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:

 [1] Bannai, E.; Bannai, Et., Generalized generalized spin models (four weight spin models), Pacific J. Math., 170, 1-16, (1995) · Zbl 0848.05072 [2] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, London, 1984. · Zbl 0555.05019 [3] Bannai, E.; Jaeger, F.; Sali, A., Classification of small spin models, Kyushu J. Math., 48, 185-200, (1994) · Zbl 0823.05065 [4] A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989. [5] Caughman, J. S., The Terwilliger algebras of bipartite P- and Q-polynomial schemes, Discrete Math., 196, 65-95, (1999) · Zbl 0924.05067 [6] Curtin, B., 2-homogeneous bipartite distance-regular graphs, Discrete Math., 187, 39-70, (1998) · Zbl 0958.05143 [7] Curtin, B., Distance-regular graphs which support a spin model are thin, Discrete Math., 197/198, 205-216, (1999) · Zbl 0929.05095 [8] Curtin, B.; Nomura, K., Some formulas for spin models on distance-regular graphs, J. Combin. Theory Ser. B, 75, 206-236, (1999) · Zbl 0930.05101 [9] Curtin, B.; Nomura, K., Spin models and strongly hyper-self-dual Bose-Mesner algebras, J. Algebraic Combin., 13, 173-186, (2001) · Zbl 0979.05111 [10] C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York, 1962. [11] Egawa, Y., Characterization of H(n, q) by the parameters, J. Combin. Theory Ser. A, 31, 108-125, (1981) · Zbl 0472.05056 [12] Jaeger, F., Towards a classification of spin models in terms of association schemes, Adv. Stud. Pure Math., 24, 197-225, (1996) · Zbl 0864.05089 [13] Jaeger, F.; Matsumoto, M.; Nomura, K., Bose-Mesner algebras related to type II matricers and spin models, J. Algebraic Combin., 8, 39-72, (1998) · Zbl 0974.05084 [14] A. Munemasa, Personal communication, 1994. [15] K. Nomura, “Spin models and almost bipartite 2-homogeneous graphs,” in Progress in Algebraic Combinatorics (Fukuoka, 1993), vol. 24 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo (1996) pp. 285-308. · Zbl 0858.05101 [16] Tanabe, K., The irreducible modules of the Terwilliger algebras of Doob schemes, J. Algebraic Combin, 6, 173-195, (1997) · Zbl 0868.05056 [17] Terwilliger, P., The subconstituent algebra of an association scheme, I, J. Algebraic Combin., 1, 363-388, (1992) · Zbl 0785.05089 [18] Yamazaki, N., Bipartite distance-regular graphs with an eigenvalue of multiplicity k, J. Combin. Theory Ser. B, 66, 34-37, (1996) · Zbl 0835.05087
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