Duality Maps of Finite Abelian Groups and Their Applications to Spin Models
Etsuko Bannai
and Akihiro Munemasa
DOI: 10.1023/A:1008613331395
Abstract
Duality maps of finite abelian groups are classified. As a corollary, spin models on finite abelian groups which arise from the solutions of the modular invariance equations are determined as tensor products of indecomposable spin models. We also classify finite abelian groups whose Bose-Mesner algebra can be generated by a spin model.
Pages: 223–233
Keywords: spin model; finite abelian group; quadratic form; association scheme
Full Text: PDF
References
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2. E. Bannai and E. Bannai, “Generalized spin models and association schemes,” Memoirs Fac. Sci. Kyushu Univ. Ser. A 47 (1993), 397-409.
3. E. Bannai and E. Bannai, “Spin models on finite cyclic groups,” J. Alg. Combin. 3 (1994), 243-259.
4. E. Bannai, E. Bannai, and F. Jaeger, “On spin models, modular invariance and duality,” J. Alg. Combin. (to appear).
5. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, Berlin-Heidelberg, 1989.
6. F. Jaeger, “Strongly regular graphs and spin models for the Kauffman polynomial,” Geom. Ded. 44 (1992), 23-52.
7. F. Jaeger, “On spin models, triply regular association schemes, and duality,” J. Alg. Combin. (to appear).
8. F. Jaeger, M. Matsumoto, and K. Nomura, “Association schemes related with type II matrices and spin models,” J. Alg. Combin. 8 (1998), 39-72.
9. V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pac. J. Math. 137 (1989), 311-336.
10. V. Kac and M. Wakimoto, “A construction of generalized spin models,” Proc. Conf. Math. Phys. (1994), 131-156.
11. Y. Kawada, “ \ddot Uber den Dualit\ddot atssatz der Charaktere nichtcommutativer Gruppen,” Proc. Phys. Math. Soc. Japan, 24(3), (1942), 97-109.
12. K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. Knot Theory and Its Ramifications 3 (1994), 465-475.
13. Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge University Press, 1993.