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Bose-Mesner algebras related to type II matrices and spin models. (English) Zbl 0974.05084

Summary: A type II matrix is a square matrix \(W\) with non-zero complex entries such that the entrywise quotient of any two distinct rows of \(W\) sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix \(W\), of a Bose-Mesner algebra \(N(W)\), which is a commutative algebra of matrices containing the identity \(I\), the all-one matrix \(J\), closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, if \(W\) is a spin model, it belongs to \(N(W)\). The transposition of matrices \(W\) corresponds to a classical notion of duality for the corresponding Bose-Mesner algebras \(N(W)\). Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute \(N(W)\) for a number of examples.

MSC:

05E30 Association schemes, strongly regular graphs
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[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 à vertex,” preprint.
[2] Ei. Bannai, “Association schemes and fusion algebras: An introduction,” J. Alg. Combin.2 (1993), 327-344. · Zbl 0790.05098
[3] Ei. Bannai and Et. Bannai, “Spin models on finite cyclic groups,” J. Alg. Combin.3 (1994), 243-259. · Zbl 0803.05058
[4] Ei. Bannai, Et. Bannai, and F. Jaeger, “On spin models, modular invariance, and duality,” preprint. · Zbl 0880.05083
[5] Ei. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984. · Zbl 0555.05019
[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. · Zbl 0747.05073
[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. · Zbl 0089.15002
[9] P.J. Cameron and J.H. van Lint, Graphs, Codes and Designs, London Math. Soc. Lecture Notes 43, Cambridge, 1980. · Zbl 0427.05001
[10] P. Delsarte, “An algebraic approach to the association schemes of coding theory,” Philips Research Reports Supplements10 (1973). · Zbl 1075.05606
[11] C.D. Godsil, Algebraic Combinatorics,Chapman and Hall, 1993. · Zbl 0784.05001
[12] F. Goodman, P. de la Harpe, and V.F.R. Jones, Coxeter Graphs and Towers of Algebras, Springer, 1989. · Zbl 0698.46050
[13] P. de la Harpe, “Spin models for link polynomials, strongly regular graphs and Jaeger”s Higman-Sims model,“ <Emphasis Type=”Italic“>Pacific J. of Math <Emphasis Type=”Bold”>162 (1994), 57-96. · Zbl 0795.57002
[14] P. de la Harpe and V.F.R. Jones, “Paires de sous-algèbres semi-simples et graphes fortement réguliers,” C.R. Acad. Sci. Paris311, Série I, (1990), 147-150. · Zbl 0707.46039
[15] P. de la Harpe and V.F.R. Jones, “Graph invariants related to statistical mechanical models: Examples and problems,” J. Combin. Theory Ser. B57 (1993) 207-227. · Zbl 0729.57003
[16] A.A. Ivanov and C.E. Praeger, “Problem session at ALCOM-91,” Europ. J. Combinatorics15 (1994), 105-112. · Zbl 0789.05013
[17] F. Jaeger, “Strongly regular graphs and spin models for the Kauffman polynomial,” Geom. Dedicata44 (1992), 23-52. · Zbl 0773.57005
[18] F. Jaeger, “On spin models, triply regular association schemes, and duality,” J. Alg. Combin.4 (1995), 103-144. · Zbl 0820.05063
[19] F. Jaeger, “New constructions of models for link invariants,” Pac. J. Math., to appear. · Zbl 0878.57007
[20] F. Jaeger, “Towards a classification of spin models in terms of association schemes,” Advanced Studies in Pure Math.24 (1996), 197-225. · Zbl 0864.05089
[21] V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pac. J. Math.137 (1989), 311-336. · Zbl 0695.46029
[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. · Zbl 0537.17001
[25] L.H. Kauffman, “An invariant of regular isotopy,” Trans. AMS318 (1990), 417-471. · Zbl 0763.57004
[26] K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. of Knot Theory and its Ramifications3 (1994), 465-475. · Zbl 0842.57010
[27] A.I. Kostrikin and P.H. Tiep, Orthogonal Decompositions and Integral Lattices, Expositions in Mathematics 15, De Gruyter, Berlin, New York, 1994. · Zbl 0855.11033
[28] A. Munemasa and Y. Watatani, “Paires orthogonales de sous-algèbres involutives,” C.R. Acad. Sci. Paris314 (1992), 329-331. · Zbl 0765.46042
[29] K. Nomura, “Spin models constructed from Hadamard matrices,” J. Combin. Theory Ser. A68 (1994), 251-261. · Zbl 0808.05100
[30] K. Nomura, “Twisted extensions of spin models,” J. Alg. Combin.4 (1995), 173-182. · Zbl 0827.05061
[31] K. Nomura, “An algebra associated with a spin model,” J. Alg. Combin.6 (1997), 53-58. · Zbl 0865.05077
[32] J.H. Van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992. · Zbl 0769.05001
[33] Wallis, W. D.; Street, A. P.; Wallis, J. S., Combinatorics: Room squares, sum-free sets, Hadamard matrices (1972), Berlin · Zbl 1317.05003
[34] D.M. Weichsel, “The Kronecker product of graphs,” Proc. AMS, 1962, Vol. 13, pp. 47-52. · Zbl 0102.38801
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