Association Schemes of Quadratic Forms and Symmetric Bilinear Forms
Yangxian Wang
, Chunsen Wang
, Changli Ma
and Jianmin Ma
Department of Mathematics, Hebei Teachers University, Shijiazhuang 050091, People's Republic of China
DOI: 10.1023/A:1022978613368
Abstract
Let X n and Y n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over \mathbb F q \mathbb{F}_q , respectively. The orbits of GL n( \mathbb F q \mathbb{F}_q ) on X n \times X n define an association scheme Qua( n, q). The orbits of GL n( \mathbb F q \mathbb{F}_q ) on Y n \times Y n also define an association scheme Sym( n, q). Our main results are: Qua( n, q) and Sym( n, q) are formally dual. When q is odd, Qua( n, q) and Sym( n, q) are isomorphic; Qua( n, q) and Sym( n, q) are primitive and self-dual. Next we assume that q is even. Qua( n, q) is imprimitive; when ( n, q) 
(2,2), all subschemes of Qua( n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt( n, q), the association scheme of alternating forms on V. The dual statements hold for Sym( n, q).
(2,2), all subschemes of Qua( n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt( n, q), the association scheme of alternating forms on V. The dual statements hold for Sym( n, q).Pages: 149–161
Keywords: association scheme; quadratic form; symmetric bilinear form
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References
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2. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, 1989.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Corrections and Additions to the book Distanc-Regular Graphs.
4. P.J. Cameron and J.J. Seidel, “Quadratic forms over GF(2),” Indag. Math. 35 (1973), 1-8.
5. L.E. Dickson, Linear Groups with Exposition of Galois Field Theory, Teubner, Leipig, 1900 and Dover, 1958.
6. Y. Egawa, “Association schemes of quadratic forms,” J. Combin. Th.(A) 38 (1981), 1-14.
7. C.D. Godsil, Algebraic Combinatorics, Chapman & Hall, 1993.
8. Y. Huo and Z. Wan, “Non-symmetric association schemes of symmetric matrices,” Acta Math. Appl. Sinica 9 (1993), 236-255.
9. Y. Huo and X. Zhu, “Association schemes with several classes of symmetric matrices,” Acta. Math. Appl. Sinica 10 (1987), 257-268.
10. J. Ma, “Fusion schemes of quadratic forms,” unpublished.
11. A. Munemasa, “An alternative construction of the graphs of quadratic forms in characteristic 2,” Algebra Colloquium 2(3) (1995), 275-287.
12. Z. Wan, Geometry of Classical Groups over Finite Fields, Studentlitteratur, Lund, 1993.
13. Y. Wang and J. Ma, “Association schemes of symmetric matrices over a finite field of characteristic two,” J. Statis. Plan and Infer. 51 (1996), 351-371.
14. Y. Wang, C. Wang, and C. Ma, “Association schemes of quadratic forms over a finite field of characteristic two,” Chinese Science Bulletin 43(23) (1998), 1965-1968.