Modular Adjacency Algebras of Hamming Schemes
Masayoshi Yoshikawa
DOI: 10.1023/B:JACO.0000048521.68503.2b
Abstract
To each association scheme G and to each field R, there is associated naturally an associative algebra, the so-called adjacency algebra RG of G over R. It is well-known that RG is semisimple if R has characteristic 0. However, little is known if R has positive characteristic. In the present paper, we focus on this case. We describe the algebra RG if G is a Hamming scheme (and R a field of positive characteristic). In particular, we show that, in this case, RG is a factor algebra of a polynomial ring by a monomial ideal.
Pages: 331–340
Keywords: association scheme; Hamming scheme; modular adjacency algebra
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. P.-J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994.
4. A. Hanaki, “Locality of a modular adjacency algebra of an association scheme of prime power order,” Arch. Math (to appear).
5. A. Hanaki, “Semisimplicity of adjacency algebras of association schemes,” J. Alg. 225 (2000), 124-129.
6. P.-H. Zieschang, An Algebraic Approach to Association Schemes, Lecture Notes in Math. vol. 1628, Springer, Berlin-Heidelberg-New York, 1996.