Commutative Association Schemes Whose Symmetrizations Have Two Classes
Sung Y. Song
DOI: 10.1023/A:1022488330352
Abstract
If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes. In particular, we give a classification of association schemes whose symmetrizations are obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes. Also the cyclotomic schemes associated to Paley graphs and their symmetrizable fission schemes are discussed in terms of their character tables.
Pages: 47–55
Keywords: cyclotomic association scheme; strongly regular graph; wreath product
Full Text: PDF
References
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2. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, California, 1984.
3. E. Bannai and S.Y. Song, "Character tables of fission schemes and fusion schemes," European J. of Combinatorics 14 (1993), 385-396.
4. E. Bannai and A. Munemasa, "Davenport-Hasse theorem and cyclotomic association schemes," Proceedings of Algebraic Combinatorics Conference, Hirosaki University, Japan (1990).
5. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance Regular Graphs, Springer-Verlag, Berlin, 1989.
6. I.A. Faradzev, "Cellular subrings of the symmetric square of a cellular ring of rank 3," Investigations in Algebraic Theory of Combinatorial Objects, I.A. Faradzev, A.A. Ivanov, M.H. Klin, and A.J. Woldar (Eds.), Kluwer Acad. Publ., Dordrecht, 1992.
7. I.A. Faradzev, A.V. Ivanov, and M.H. Klin, "Galois correspondence between permutation groups and cellular rings," Graphs and Combinatorics 6 (1990), 303-332.
8. I.A. Faradzev, M.H. Klin, and M.E. Muzichuk, "Cellular rings and groups of automorphisms of graphs," Investigations in Algebraic Theory of Combinatorial Objects, I.A. Faradzev, A.A. Ivanov, M.H. Klin, and A.J. Woldar (Eds.), Kluwer Acad. Publ., Dordrecht, 1992.
9. D.G. Higman, "Coherent configurations, Part I, Ordinary representation theory," Geometriae Dedicata 4 (1975), 1-32.
10. K.W. Johnson and J.D.H. Smith, "Characters of finite quasigroups III: Quotients and fusion," Europ. J. Comb. 19 (1989), 47-56.
11. M.H. Klin and R. Poschel, "The Konig problem, the isomorphism problem for cyclic groups and the method of Schur rings," Algebraic Methods in Graph Theory, North Holland, Amsterdam, pp. 405-434, 1981.
12. M.E. Muzichuk, "Subcells of the symmetric cells," Algebraic Structures and Their Applications, Kiev, 1988, 172-174, in Russian.
13. S.Y. Song, "Class 3 association schemes whose symmetrizations have two classes," J. of Combin. Theory (A) 70(1995), 1-29.
14. B. Weisfeiler, On Construction and Identification of Graphs, Springer-Verlag, Berlin, 1976.