Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X_n and Y_n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL_n() on X_n × X_n define an association scheme Qua(n, q). The orbits of GL_n() on Y_n × 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).