On Modular Standard Modules of Association Schemes
Akihide Hanaki1
and Masayoshi Yoshikawa2
1Department of Mathematical Sciences, Faculty of Science Shinshu University Matsumoto 390-8621 Japan
2Department of Mathematical Sciences, Graduate School of Science and Technology Shinshu University Matsumoto 390-8621 Japan
2Department of Mathematical Sciences, Graduate School of Science and Technology Shinshu University Matsumoto 390-8621 Japan
DOI: 10.1007/s10801-005-6911-3
Abstract
We will determine the structure of the modular standard modules of association schemes of class two. In the process, we will give the theoretical interpretation for the p-rank theory for strongly regular graphs, and understand the p-rank as the dimension of a submodule of the modular standard module. Considering the modular standard module, we can obtain the detailed classification more than the p-rank and the parameters.
Pages: 269–279
Keywords: key words association scheme; modular adjacency algebra; modular standard module; $p$-rank; strongly regular graph
Full Text: PDF
References
1. Z. Arad, E. Fisman, and M. Muzychuk, “Generalized table algebras,” Israel J. Math. 144 (1999), 29-60.
2. E. Bannai and T. Ito, “Algebraic combinatorics. I,” Association Schemes, Benjamin-Cummings, Menlo Park, CA, 1984.
3. D. Benson, Representation and Cohomology I, Cambridge, 1995.
4. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.
5. A.E. Brouwer and C.A. van Eijl, “On the p-rank of the adjacency matrices of strongly regular graphs,” J. Algebraic Combin. 1 (1992), 329-346.
6. A. Hanaki, “Semisimplicity of adjacency algebras of association schemes,” J. Algebra 225 (2000), 124-129.
7. H. Nagao and Y. Tsushima, Representations of Finite Groups, Academic Press, San Diego CA, 1987.
8. R. Peeters, “Uniqueness of strongly regular graphs having minimal p-rank,” Linear Algebra Appl. 226-228 (1995), 9-31.
9. R. Peeters, “On the p-ranks of the adjacency matrices of distance-regular graphs,” J. Algebraic Combin. 15 (2002), 127-149.
10. E. Spence, “Regular two-graphs on 36 vertices,” Linear Algebra Appl. 459-497 (1995), 226-228.
11. E. Spence, http://www.maths.gla.ac.uk/%7ees/srgraphs.html .
12. P.-H. Zieschang, An Algebraic Approach to Association Schemes, vol. 1628, Lecture Notes in Math Springer, Berlin, Heidelberg, New York, 1996.
2. E. Bannai and T. Ito, “Algebraic combinatorics. I,” Association Schemes, Benjamin-Cummings, Menlo Park, CA, 1984.
3. D. Benson, Representation and Cohomology I, Cambridge, 1995.
4. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.
5. A.E. Brouwer and C.A. van Eijl, “On the p-rank of the adjacency matrices of strongly regular graphs,” J. Algebraic Combin. 1 (1992), 329-346.
6. A. Hanaki, “Semisimplicity of adjacency algebras of association schemes,” J. Algebra 225 (2000), 124-129.
7. H. Nagao and Y. Tsushima, Representations of Finite Groups, Academic Press, San Diego CA, 1987.
8. R. Peeters, “Uniqueness of strongly regular graphs having minimal p-rank,” Linear Algebra Appl. 226-228 (1995), 9-31.
9. R. Peeters, “On the p-ranks of the adjacency matrices of distance-regular graphs,” J. Algebraic Combin. 15 (2002), 127-149.
10. E. Spence, “Regular two-graphs on 36 vertices,” Linear Algebra Appl. 459-497 (1995), 226-228.
11. E. Spence, http://www.maths.gla.ac.uk/%7ees/srgraphs.html .
12. P.-H. Zieschang, An Algebraic Approach to Association Schemes, vol. 1628, Lecture Notes in Math Springer, Berlin, Heidelberg, New York, 1996.