Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras
Brian Curtin
and Kazumasa Nomura
DOI: 10.1023/A:1011297515395
Abstract
We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let M \mathcal{M} denote a Bose-Mesner algebra on a finite nonempty set X. Fix p M * \mathcal{M}^ * and T \mathcal{T} denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of M \mathcal{M} with respect to p. By a hyper-duality of M \mathcal{M} , we mean an automorphism T \mathcal{T} such that y( M) = M * , y 2 ( A) = t A ψ(\mathcal{M}) = \mathcal{M}^ * ,ψ^2 (A) = ^t {\kern 1pt} A for all A Ĩ M A \in \mathcal{M} ; and | X | y r \left| X \right|ψρ is a duality of M \mathcal{M} . M \mathcal{M} is said to be hyper-self-dual whenever there exists a hyper-duality of M \mathcal{M} . We say that M \mathcal{M} is strongly hyper-self-dual whenever there exists a hyper-duality of M \mathcal{M} which can be expressed as conjugation by an invertible element of T \mathcal{T} . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.
Pages: 173–186
Keywords: Bose-mesner algebra; Terwilliger algebra; spin model
Full Text: PDF
References
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21. V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pac. J. Math. 137 (1989), 311-224.
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2. E. Bannai, Et. Bannai, T. Ikuta, and K. Kawagoe, “Spin models constructed from the Hamming association schemes,” in Proceedings of the 10th Algebraic Combinatorics Symposium at Gifu University, 1992.
3. E. Bannai, Et. Bannai, and F. Jaeger, “On spin models, modular invariance, and duality,” J. Alg. Combin. 6 (1997), 203-228. CURTIN AND NOMURA
4. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984.
5. E. Bannai, F. Jaeger, and A. Sali, “Classification of small spin models,” Kyushu J. of Math. 48 (1994), 185-200.
6. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, New York, 1989.
7. B. Curtin, “Distance-regular graphs which support a spin model are thin,” Discr. Math. 197/198 (1999), 205-216.
8. B. Curtin, “The Terwilliger algebra of a 2-homogeneous bipartite distance-regular graph,” J. Combin. Theory Ser. B, to appear.
9. B. Curtin, “Hyper-dual pairs of Bose-Mesner algebras, I, II,” preprint.
10. B. Curtin and K. Nomura, “Some formulas for spin models on distance-regular graphs,” J. Comb. Theory Ser. B. 75 (1999), 206-236.
11. B. Curtin and K. Nomura, “Distance-regular graphs related to the quantum enveloping algebra of sl(2), J. Alg. Comb. 12 (2000), 25-36.
12. C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York, 1962.
13. P. Delsarte, “An algebraic approach to the association schemes of coding theory,” Philips Research Reports Supplements 10 (1973).
14. M. Doob, “On graph products and association schemes,” Utilitas Math. 1 (1972), 291-302.
15. J. Go, “The Terwilliger algebra of the hypercube,” preprint.
16. C.D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
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, “Towards a classification of spin models in terms of association schemes,” Advanced Studies in Pure Math. 24 (1996), 197-225.
20. 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.
21. V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pac. J. Math. 137 (1989), 311-224.
22. K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. Knot Th. Ramif. 3 (1995), 465-475.
23. A. Munemasa, private communication.
24. A. Neumaier, “Duality in coherent configurations,” Combinatorica 9 (1989), 59-67.
25. K. Nomura, “Spin models constructed from Hadamard matrices,” J. Combin.Theory Ser. A 68 (1994), 251-261.
26. K. Nomura, “An algebra associated with a spin model,” J. Alg. Combin. 6 (1997), 53-58.
27. S.S. Shrikhande, “The uniqueness of the L2 association scheme,” Ann. Math. Statist. 30 (1959), 781-798.
28. N.J.A. Sloane, “An introduction to association schemes and coding theory,” Theory and Application of Special Functions 35 (1975), 225-260.
29. K. Tanabe, “The irreducible modules of the Terwilliger algebras of Doob schemes,” J. Alg.Combin. 6 (1997), 173-195.
30. P. Terwilliger, “The subconstituent algebra of an association scheme,” J. Alg.Combin. Part I: 1 (1992), 363-388; Part II: 2 (1993), 73-103; Part III: 2 (1993), 177-210.
31. M. Tomiyama and N. Yamazaki, “The Terwilliger algebra of a strongly regular graph,” Kyushu J. Math 48 (1994), 323-334.