Type II Matrices and Their Bose-Mesner Algebras
Rie Hosoya
and Hiroshi Suzuki
DOI: 10.1023/A:1021960623533
Abstract
Type II matrices were introduced in connection with spin models for link invariants. It is known that a pair of Bose-Mesner algebras (called a dual pair) of commutative association schemes are naturally associated with each type II matrix. In this paper, we show that type II matrices whose Bose-Mesner algebras are imprimitive are expressed as so-called generalized tensor products of some type II matrices of smaller sizes. As an application, we give a classification of type II matrices of size at most 10 except 9 by using the classification of commutative association schemes.
Pages: 19–37
Keywords: type II matrix; spin model; Bose-mesner algebra
Full Text: PDF
References
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2. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin-Cummings, California, 1984.
3. E. Bannai and M. Sawano, “The classification of spin models of certain four-weight spin models,” Ann. Comb. 4(2) (2000), 139-151.
4. H. Guo and T. Huang, “Some classes of four-weight spin models,” in Second Shanghai Conference on Designs, Codes and Finite Geometries,
1996. Also in J. Statist. Plann. Inference, 94(2) (2001), 231-247.
5. U. Haagerup, “Orthogonal maximal Abelian *-subalgebras of the n \times n matrices and cyclic n-roots,” in Operator Algebras and Quantum Field Theory, S. Doplicher, R. Longo, J.E. Roberts, and L. Zsido (Eds.), International Press, 1998.
6. F. Jaeger, “On four-weight spin models and their gauge transformation,” J. Alg. Comb. 11 (2000), 241-268.
7. F. Jaeger, M. Matsumoto, and K. Nomura, “Bose-Mesner algebras related to type II matrices and spin models,” J. Alg. Comb. 8 (1998), 39-72.
8. V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pacific Journal of Mathematics, 137 (1989), 311-334.
9. K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. Knot Theory and Its Ramification 3 (1994), 465-476.
10. T. Matsumura, “Bose-Mesner algebras over K and their related type II matrices,” Master's Thesis, International Christian University, 2001.
11. E. Nomiyama, “Classification of association schemes with at most ten vertices,” Kyushu J. Math. 49 (1995), 163-195.
12. K. Nomura, “Twisted extensions of spin models,” J. Alg. Combin. 4 (1995), 173-182.
13. K. Nomura, “An algebra associated with a spin model,” J. Alg. Combin. 6 (1997), 53-58.
14. K. Nomura, “Type II matrices of size five,” Graphs and Combinatorics, 15 (1999), 79-92.
15. H. Suzuki and H. Tsuchiyama, “A classification of four-weight spin models of size six and seven,” Preprint.