Algebraic structure of association schemes of prime order
Akihide Hanaki1
and Katsuhiro Uno2
1Shinshu University Faculty of Science Matsumoto 390-8621 Japan
2Osaka Kyoiku University Department of Mathematical Sciences Kashiwara Osaka 582-8582 Japan
2Osaka Kyoiku University Department of Mathematical Sciences Kashiwara Osaka 582-8582 Japan
DOI: 10.1007/s10801-006-6923-7
Abstract
Finite groups of prime order must be cyclic. It is natural to ask what about association schemes of prime order. In this paper, we will give an answer to this question. An association scheme of prime order is commutative, and its valencies of nontrivial relations and multiplicities of nontrivial irreducible characters are constant. Moreover, if we suppose that the minimal splitting field is an abelian extension of the field of rational numbers, then the character table is the same as that of a Schurian scheme.
Pages: 189–195
Keywords: keywords association scheme; prime order; cyclotomic scheme; character
Full Text: PDF
References
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2. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin-Cummings, Menlo Park CA, 1984.
3. J.S. Frame, “The double cosets of a finite group,” Bull. Amer. Math. Soc. 47 (1941), 458-467.
4. A. Hanaki, “Semisimplicity of adjacency algebras of association schemes,” J. Algebra 225 (2000), 124-129.
5. A. Hanaki, “Locality of a modular adjacency algebra of an association scheme of prime power order,” Arch. Math. 79 (2002), 167-170.
6. D. G. Higman, “Schur relations for weighted adjacency algebras,” Symp. Math. Roma (London-New York), 13 (1974), 467-477.
7. A. Munemasa, “Splitting fields of association schemes,” J. Combin. Theory. Ser. A 57 (1991), 157-161.
8. W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 3rd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 2004.
9. P.-H. Zieschang, “An algebraic approach to association schemes,” Lecture Notes in Mathematics, 1628, Springer-Verlag, Berlin, 1996.