Association Schemes and Fusion Algebras (An Introduction)
Eiichi Bannai
DOI: 10.1023/A:1022489416433
Abstract
We introduce the concept of fusion algebras at algebraic level, as a purely algebraic concept for the fusion algebras which appear in conformal field theory in mathematical physics. We first discuss the connection between fusion algebras at algebraic level and character algebras, a purely algebraic concept for Bose-Mesner algebras of association schemes. Through this correspondence, we establish the condition when the matrix S of a fusion algebra at algebraic level is unitary or symmetric. We construct integral fusion algebras at algebraic level, from association schemes, in particular from group association schemes, whose matrix S is unitary and symmetric. Finally, we consider whether the modular invariance property is satisfied or not, namely whether there exists a diagonal matrix T satisfying the condition ( ST) 3 = S 2. We prove that this property does not hold for some integral fusion algebras at algebraic level coming from the group association scheme of certain groups of order 64, and we also prove that the (nonintegral) fusion algebra at algebraic level obtained from the Hamming association scheme H( d, q) has the modular invariance property.
Pages: 327–344
Keywords: association scheme; fusion algebra; character algebra; Bose-mesner algebra; modular invariance property
Full Text: PDF
References
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2. E. Bannai and E. Bannai, "Modular invariance of the character table of the Hamming association scheme H(d, q)," J, of Number Theory, to appear.
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2. E. Bannai and E. Bannai, "Modular invariance of the character table of the Hamming association scheme H(d, q)," J, of Number Theory, to appear.
3. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, CA, 1984.
4. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance Regular Graphs, Springer-Verlag, 1989.
5. P. Delsarte, "An algebraic approach to the association schemes of coding theory," Philips Res. Reports Suppls 10, (1973).
6. R. Dijkgraaf and E. Verlinde, "Modular invariance and the fusion algebra," Nucl. Phys. B (Proc. suppl.) S (1988), 87-97.
7. R. Dijkgraaf, C. Vafa, E. Verlinde, and H. Verlinde, "The operator algebra of orbifold models," Comm. Math. Phys. 123 (1989), 485-526.
8. M. Hall, Jr. and J.K. Senior, The groups of order 2n(n < 6), Macmillan Company, New York, NY, 1964.
9. Y. Kawada, "Uber den dualitatssatz der Charactere nichtcommutativer Gruppen," Proc. Math.- Phys. Soc. Japan (3)24 (1942), 97-109.
10. T. Kohno, "Topological quantum field theory, -centering around the application to 3-dimensional manifolds" Sugaku 44 (1992), 29-43 (in Japanese).
11. G. Lusztig, "Leading coefficients of character values of Hecke algebras," Proc. Symp. Pure Math. 47 (1987), 253-262.
12. C. Moore and N. Seiberg, "Classical and quantum conformal field theory," Comm. Math. Phys. 123 (1989), 177-254.
13. E. Verlinde, "Fusion rules and modular transformations in 2-dimensional conformal field theories," Nucl. Phys. B. 300 (1988), 360-376.