Spin Models on Finite Cyclic Groups
Eiichi Bannai
and Etsuko Bannai
DOI: 10.1023/A:1022407800541
Abstract
The concept of spin model is due to V. F. R. Jones. The concept of nonsymmetric spin model, which generalizes that of the original (symmetric) spin model, is defined naturally. In this paper, we first determine the diagonal matrices T satisfying the modular invariance or the quasi modular invariance property, i.e., ( PT) 3 = Ö{ mP 2 } (PT)^3 = \sqrt {mP^2 } or ( PT) 3 = m \frac32 I (PT)^3 = m^{\frac{3}{2}} I (respectively), for the character table P of the group association scheme of a cyclic group G of order m. Then we show that a (symmetric or nonsymmetric) spin model on G is constructed from each of the matrices T satisfying the modular or quasi modular invariance property.
Pages: 243–259
Keywords: spin model; association scheme; cyclic group; modular invariance property; link invariant
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References
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2. Bannai, E. and Bannai, E., "Generalized generalized spin models (four-weight spin models)," to appear in Pac. J. Math.
3. Bannai, E. and Ito, T., Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park CA, 1984. 4. de la Harpe, P., Spin models for link polynomials, strongly regular graphs and Jaeger's Higman-Sims model, preprint.
5. Fateev, V.A. and Zamolodchikov, A. B., "Self-dual solutions of the star-triangle relations in ZN models," Phys. Lett. A 92 (1982), 37-39.
6. Ooldschmidt, D. M. and Jones, V.F.R., "Metaplectic link invariants," Geom. Dedicate 31 (1989), 165-191.
7. Ireland, K. and Rosen, M.,A Classical Introduction to Modem Number Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982.
8. Jaeger, F., "Strongly regular graphs and spin models for the Kauffman polynomial," Geom. Dedicata 44 (1992), 23-52.
9. Jones, V.F.R., "On knot invariants related to some statistical mechanical models," Pac. J. Math. 137 (1989), 311-334.
10. Kauffman, L.H., "An invariant of regular isotopy," Trans. Amer. Math. Soc. 318(2) (1990), 417-471.
11. Kobayashi, T., Murakami, H. and Murakami, J., "Cyclotomic invariants for links," Proc. Japan Acad. 64A (1988), 235-238.
12. Kashiwara, M. and Miwa, T., "A class of elliptic solutions to the star-triangle relation," Nuclear Physics B275 [FS17] (1988) 121-134.
13. Kawagoe, K., Munemasa, A., and Watatani, Y., Generalized spin models, preprint.
14. Nagell, T., Introduction to Number Theory, Almqvist and Wiksell, Stockholm, and John Wiley and Sons, New York, 1951 (Reprinted by Chelsea Publishing Company, New York.).
15. Schur, I., Qber die Gaupschen Summen, GesammelleAbhandlungen, vol II, Springer-Verlag, 1973,327-333.