A modular absolute bound condition for primitive association schemes
Akihide Hanaki1
and Ilia Ponomarenko2
1Shinshu University Faculty of Science Matsumoto 390-8621 Japan
2Petersburg Department of V.A.Steklov Institute of Mathematics St. Petersburg 191023 Russia
2Petersburg Department of V.A.Steklov Institute of Mathematics St. Petersburg 191023 Russia
DOI: 10.1007/s10801-008-0145-0
Abstract
The well-known absolute bound condition for a primitive symmetric association scheme ( X, S) gives an upper bound for | X| in terms of | S| and the minimal non-principal multiplicity of the scheme. In this paper we prove another upper bounds for | X| for an arbitrary primitive scheme ( X, S). They do not depend on | S| but depend on some invariants of its adjacency algebra KS where K is an algebraic number field or a finite field.
Pages: 447–456
Keywords: keywords association scheme; adjacency algebra
Full Text: PDF
References
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2. Dade, E.C.: Block extensions. Ill. J. Math. 17, 198-272 (1973)
3. Evdokimov, S.A.: Schurity and separability of associative schemes. Doctor of Sciences Thesis, St. Petersburg State University (2005)
4. Evdokimov, S., Ponomarenko, I.: Two inequalities for the parameters of a cellular algebra. Zap. Nauchnykh Seminarov POMI 240, 82-95 (1997). English translation: J. Math. Sci., New York 96(5), 3496-3504 (1999)
5. Hanaki, A.: Semisimplicity of adjacency algebras of association schemes. J. Algebra 225, 124-129 (2000)
6. Hanaki, A., Yoshikawa, M.: On modular standard modules of association schemes. J. Algebraic Combin. 21, 269-279 (2005)
7. Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers. Springer Monographs in Mathematics. Springer, Berlin (2004)
8. Peeters, R.: On the p-ranks of the adjacency matrices of distance-regular graphs. J. Algebraic Combin.