Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let Xn and Yn be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over \Bbb fq, respectively. The orbits of GLn(\Bbb fq) on Xn × Xn define an association scheme Qua(n, q). The orbits of GLn(\Bbb fq) on Yn × Yn 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 isomorphics Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitives 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).