General Form of Non-Symmetric Spin Models
Takuya Ikuta
and Kazumasa Nomura
DOI: 10.1023/A:1008711618027
Abstract
A spin model (for link invariants) is a square matrix W with non-zero complex entries which satisfies certain axioms. Recently (Jaeger and Nomura, J. Alg. Combin. 10 (1999), 241-278) it was shown that t WW -1 is a permutation matrix (the order of this permutation matrix is called the 
index 
of W), and a general form was given for spin models of index 2. In the present paper, we generalize this general form to an arbitrary index m. In particular, we give a simple form of W when m is a prime number.
index of W), and a general form was given for spin models of index 2. In the present paper, we generalize this general form to an arbitrary index m. In particular, we give a simple form of W when m is a prime number.Pages: 59–72
Keywords: spin model; association scheme; Bose-mesner algebra
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References
1. Et. Bannai, “Modular invariance property and spin models attached to cyclic group association schemes,” J. Stat. Plann. and Inference 51 (1996), 107-124.
2. E. Bannai and Et. Bannai, “Spin models on finite cyclic groups,” J. Alg. Combin. 3 (1994), 243-259.
3. E. Bannai, Et. Bannai, and F. Jaeger, “On spin models, modular invariance, and duality,” J. Alg. Combin. 6 (1997), 203-228.
4. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984.
5. F. Jaeger, M. Matsumoto, and K. Nomura, “Bose-Mesner algebras related to type II matrices and spin models,” J. Alg. Combin. 8 (1998), 39-72.
6. F. Jaeger and K. Nomura, “Symmetric versus non-symmetric spin models for link invariants,” J. Alg. Combin. 10 (1999), 241-278.
7. V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pac. J. Math. 137 (1989), 311-336.
8. K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. of Knot Theory and its Ramifications 3 (1994), 465-475.
2. E. Bannai and Et. Bannai, “Spin models on finite cyclic groups,” J. Alg. Combin. 3 (1994), 243-259.
3. E. Bannai, Et. Bannai, and F. Jaeger, “On spin models, modular invariance, and duality,” J. Alg. Combin. 6 (1997), 203-228.
4. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984.
5. F. Jaeger, M. Matsumoto, and K. Nomura, “Bose-Mesner algebras related to type II matrices and spin models,” J. Alg. Combin. 8 (1998), 39-72.
6. F. Jaeger and K. Nomura, “Symmetric versus non-symmetric spin models for link invariants,” J. Alg. Combin. 10 (1999), 241-278.
7. V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pac. J. Math. 137 (1989), 311-336.
8. K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. of Knot Theory and its Ramifications 3 (1994), 465-475.