Bose-Mesner Algebras Related to Type II Matrices and Spin Models

Journal of Algebraic Combinatorics

Volumn 8, Issue 1, 1998, Pages 39-72

  Jaeger, François ^a   Matsumoto, Makoto ^b   Nomura, Kazumasa ^c  

^a Institut IMAG   (France)

^b KEIO UNIVERSITY   (Japan)

^c TOKYO MEDICAL AND DENTAL UNIVERSITY   (Japan)

Author keywords

Association scheme; Bose Mesner algebra; Link invariant; Spin model

Indexed keywords

EID: 0002919154     PISSN: 09259899     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008691327727     Document Type: Article

References (34)

1. R. Bacher, P. de la Harpe, and V.F.R. Jones, "Tours de centralisateurs pour les paires d'algèbres, modèles à spins et modèles àvertex," preprint.
2. Ei. Bannai, "Association schemes and fusion algebras: An introduction," J. Alg. Combin. 2 (1993), 327-344.
3. Ei. Bannai and Et. Bannai, "Spin models on finite cyclic groups," J. Alg. Combin. 3 (1994), 243-259.
4. Ei. Bannai, Et. Bannai, and F. Jaeger, "On spin models, modular invariance, and duality," preprint.
5. Ei. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984.
6. Et. Bannai and A. Munemasa, "Duality maps of finite abelian groups and their applications to spin models," preprint, 1995.
7. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.
8. R.C. Bose and D.M. Mesner, "On linear associative algebras corresponding to association schemes of partially balanced designs," Ann. Math. Statist. 30 (1959), 21-38.
9. P.J. Cameron and J.H. van Lint, Graphs, Codes and Designs, London Math. Soc. Lecture Notes 43, Cambridge, 1980.
10. P. Delsarte, "An algebraic approach to the association schemes of coding theory," Philips Research Reports Supplements 10 (1973).
11. C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, 1993.
12. F. Goodman, P. de la Harpe, and V.F.R. Jones, Coxeter Graphs and Towers of Algebras, Springer, 1989.
13. P. de la Harpe, "Spin models for link polynomials, strongly regular graphs and Jaeger's Higman-Sims model," Pacific J. of Math 162 (1994), 57-96.
14. P. de la Harpe and V.F.R. Jones, "Paires de sous-algèbres semi-simples et graphes fortement réguliers," C.R. Acad. Sci. Paris 311, Série I, (1990), 147-150.
15. P. de la Harpe and V.F.R. Jones, "Graph invariants related to statistical mechanical models: Examples and problems," J. Combin. Theory Ser. B 57 (1993) 207-227.
16. A.A. Ivanov and C.E. Praeger, "Problem session at ALCOM-91," Europ. J. Combinatorics 15 (1994), 105-112.
17. F. Jaeger, "Strongly regular graphs and spin models for the Kauffman polynomial," Geom. Dedicata 44 (1992), 23-52.
18. F. Jaeger, "On spin models, triply regular association schemes, and duality," J. Alg. Combin. 4 (1995), 103-144.
19. F. Jaeger, "New constructions of models for link invariants," Pac. J. Math., to appear.
20. F. Jaeger, "Towards a classification of spin models in terms of association schemes," Advanced Studies in Pure Math. 24 (1996), 197-225.
21. V.F.R. Jones, "On knot invariants related to some statistical mechanical models," Pac. J. Math. 137 (1989), 311-336.
22. V.F.R. Jones, private communication.
23. V.F.R. Jones and V.S. Sunder, Introduction to Subfactors, to appear.
24. V.G. Kac, Infinite Dimensional Lie Algebras, Progress in Mathematics 44, Birkhäusen, Boston, Basel, Stuttgart, 1983.
25. L.H. Kauffman, "An invariant of regular isotopy," Trans. AMS 318 (1990), 417-471.
26. K. Kawagoe, A. Munemasa, and Y. Watatani, "Generalized spin models," J. of Knot Theory and its Ramifications 3 (1994), 465-475.
27. A.I. Kostrikin and P.H. Tiep, Orthogonal Decompositions and Integral Lattices, Expositions in Mathematics 15, De Gruyter, Berlin, New York, 1994.
28. A. Munemasa and Y Watatani, "Paires orthogonales de sous-algèbres involutives," C.R. Acad. Sci. Paris 314 (1992), 329-331.
29. K. Nomura, "Spin models constructed from Hadamard matrices," J. Combin. Theory Ser. A 68 (1994), 251-261.
30. K. Nomura, "Twisted extensions of spin models," J. Alg. Combin. 4 (1995), 173-182.
31. K. Nomura, "An algebra associated with a spin model," J. Alg. Combin. 6 (1997), 53-58.
32. J.H. Van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992.
33. W.D. Wallis, A.P. Street, and J.S. Wallis, "Combinatorics: Room squares, sum-free sets, Hadamard matrices," Lecture Notes in Math. 292, Springer-Verlag, Berlin, 1972.
34. D.M. Weichsel, "The Kronecker product of graphs," Proc. AMS, 1962, Vol. 13, pp. 47-52.