Semidirect Products of Association Schemes
Sejeong Bang1
, Mitsugu Hirasaka2
and Sung-Yell Song3
1Combinatorial and Computational Mathematics Center Pohang University of Science and Technology Pohang 790-784 Korea
2Department of Mathematics Pusan National University Pusan 609-735 Korea
3Department of Mathematics Iowa State University Ames Iowa 50011 USA
2Department of Mathematics Pusan National University Pusan 609-735 Korea
3Department of Mathematics Iowa State University Ames Iowa 50011 USA
DOI: 10.1007/s10801-005-6278-5
Abstract
In his 1996 work developing the theory of association schemes as a 
generalized 
group theory, Zieschang introduced the concept of the semidirect product as a possible product operation of certain association schemes. In this paper we extend the semidirect product operation into the entire set of association schemes. We then derive a way to decompose certain association schemes into smaller association schemes. We also investigate to what extent this product helps us to understand and characterize the structure of association schemes. We give some examples to show that the semidirect product produces many schemes that cannot be described as neither the direct product nor the wreath product of smaller schemes.
generalized group theory, Zieschang introduced the concept of the semidirect product as a possible product operation of certain association schemes. In this paper we extend the semidirect product operation into the entire set of association schemes. We then derive a way to decompose certain association schemes into smaller association schemes. We also investigate to what extent this product helps us to understand and characterize the structure of association schemes. We give some examples to show that the semidirect product produces many schemes that cannot be described as neither the direct product nor the wreath product of smaller schemes.Pages: 23–38
Keywords: keywords semidirect product; association scheme
Full Text: PDF
References
1. S. Bang and S.Y. Song, “Characterization of maximal rational circulant association schemes,” in Codes and Designs, K.T. Arasu and A. Seress (Eds.), Ohio State Univ. Math. Inst. Publ. 10, Walter de Gruyter, Berlin, 2002, pp. 37-48.
2. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, CA, 1984.
3. P.J. Cameron, Permutation Groups, Cambridge University Press, Cambridge, UK, 1999.
4. A. Hanaki and I. Miyamoto, “Classification of primitive association schemes of order up to 22,” Kyushu J. Math. 54(1) (2000), 81-86. (cf. http://kissme.shinshu-u.ac.jp/as/)
5. M. Hirasaka, “The classification of association schemes with 11 or 12 vertices,” Kyushu J. Math. 51 (1997), 413-428.
6. M. Muzychuk, Semidirect product, a short note, March 22, 2000.
7. M. Muychuk, M. Klin, and R. P\ddot oschel, “The isomorphism problem for circulant graphs via Schur ring theory,” DIMACS Series in Discrete Mathematics 56 (2001), 241-264.
8. K. See and S. Y. Song, “Association schemes of small order,” J. Statist. Plann. Inference 73 (1998), 225-271.
9. H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
10. P.-H. Zieschang, An Algebraic Approach to Association Schemes, Lecture Notes in Mathematics, Vol. 1628, Springer, 1996.
2. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, CA, 1984.
3. P.J. Cameron, Permutation Groups, Cambridge University Press, Cambridge, UK, 1999.
4. A. Hanaki and I. Miyamoto, “Classification of primitive association schemes of order up to 22,” Kyushu J. Math. 54(1) (2000), 81-86. (cf. http://kissme.shinshu-u.ac.jp/as/)
5. M. Hirasaka, “The classification of association schemes with 11 or 12 vertices,” Kyushu J. Math. 51 (1997), 413-428.
6. M. Muzychuk, Semidirect product, a short note, March 22, 2000.
7. M. Muychuk, M. Klin, and R. P\ddot oschel, “The isomorphism problem for circulant graphs via Schur ring theory,” DIMACS Series in Discrete Mathematics 56 (2001), 241-264.
8. K. See and S. Y. Song, “Association schemes of small order,” J. Statist. Plann. Inference 73 (1998), 225-271.
9. H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
10. P.-H. Zieschang, An Algebraic Approach to Association Schemes, Lecture Notes in Mathematics, Vol. 1628, Springer, 1996.