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Annals of Combinatorics 4
(2000) 139-151
The Classification of Certain Four-Weight Spin Models
Etsuko Bannai and Mitsuhiro Sawano
Graduate School of Mathematics, Kyushu University, 6-10-1 Hakozaki,
Higashi-ku, Fukuoka 812-8581, Japan
etsuko@math.kyushu-u.ac.jp
Received March 27, 1999
**AMS Subject Classification**: 05E30,
05E99
**Abstract.** In this paper we show
that if one of the matrices
of a four-weight spin model
is equivalent to the matrix of a Potts model or a cyclic model as type
II matrix and ,
then the spin model is gauge equivalent to a Potts model or a cyclic model
up to simultaneous permutations on rows and columns. Using this fact and
Nomura's result [12] we show that every four-weight spin model of size
is gauge equivalent to either a Potts model or a cyclic model up to simultaneous
permutations on rows and columns.
**Keywords**: four-weight spin models,
Bose-Mesner algebra, gauge equivalent
**References**
1. Et. Bannai, *Modular invariance property and spin models attached to cyclic group association
schemes*, J. Statistical Planning and Inference 51 (1996) 107–124.
2. Ei. Bannai and Et. Bannai, *Generalized generalized spin models *(*four-weight spin models*),
Pacific J. Math. 170 (1995) 1–16.
3. Ei. Bannai, Et. Bannai, and F. Jaeger, *On spin models*, *modular invariance*, *and duality*, J.
Alg. Combin. 6 (1997) 203–228.
4. T. Deguchi, *Generalized spin models associated with exactly solvable models*, In: Progress
in Algebraic Combinatorics, Advanced Studies in Pure Mathematics, Vol. 24, Mathematical
Society of Japan, 1996, pp. 81–100.
5. H. Guo, *On four-weight spin models*, Ph.D. Thesis, Kyushu University, 1997.
6. H. Guo and T. Huang, *Four-weight spin models and related Bose–Mesner algebras*, preprint.
7. H. Guo and T. Huang, *Some classes of four-weight spin models*, preprint.
8. F. Jaeger,* On four-weight spin models and their gauge transformation*, J. Alg. Combin., to
appear.
9. 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.
10. V.F.R. Jones, *On knot invariants related to some statistical mechanical models*, Pacific J.
Math. 137 (1989) 311–334.
11. K. Kawagoe, A. Munemasa, and Y. Watatani, *Generalized spin models*, J. Knot Theory and
its Ramifications 3 (1994) 465–476.
12. K. Nomura,* Type II matrices of size five*, Graphs and Combin. 15 (1999) 79–92.
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