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

Received March 27, 1999

AMS Subject Classification: 05E30, 05E99

Abstract. In this paper we show that if one of the matrices $\{W_i,\ 1\leq i\leq 4 \}$ of a four-weight spin model (X,W_1,W_2,W_3,W_4;\,D) is equivalent to the matrix of a Potts model or a cyclic model as type II matrix and $|X|\geq 5$, 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 |X|=5 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


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.

Get the DVI | PS | PDF file of this abstract.