Spin Models of Index 2 and Hadamard Models
Kazumasa Nomura
DOI: 10.1023/A:1021952206694
Abstract
A spin model (for link invariants) is a square matrix W with non-zero complex entries which satisfies certain axioms. Recently 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. Moreover, new spin models, called non-symmetric Hadamard models, were constructed. In the present paper, we classify certain spin models of index 2, including non-symmetric Hadamard models.
index of W), and a general form was given for spin models of index 2. Moreover, new spin models, called non-symmetric Hadamard models, were constructed. In the present paper, we classify certain spin models of index 2, including non-symmetric Hadamard models.Pages: 5–17
Keywords: spin model; association scheme; Hadamard matrix; Potts model
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References
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2. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Reguar Graphs, Springer-Verlag, Berlin, 1989.
4. T. Ikuta and K. Nomura, “General form of non-symmetric spin models,” J. Alg. Combin. 12 (2000), 59-72.
5. F. Jaeger, “Towards a classification of spin models in terms of association schemes,” in Progress in Algebraic Combinatorics, Advanced Studies in Pure Math., Vol. 24, Math. Soc. Japan, 1996, pp. 197-225.
6. 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.
7. F. Jaeger and K. Nomura, “Symmetric versus non-symmetric spin models for link invariants,” J. Alg. Combin. 10 (1999), 241-278.
8. V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pac. J. Math. 137 (1989), 311-336.
9. K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. Knot Th. Its Ramific. 3 (1994), 465-475.
10. K. Nomura, “Spin models constructed from Hadamard matrices,” J. Combin. Th. Ser. A 68 (1994), 251-261.
11. K. Nomura, “An algebra associated with a spin model,” J. Alg. Combin. 6 (1997), 53-58.