Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

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 \(\mathbb{F}_q \), respectively. The orbits of GL n(\(\mathbb{F}_q \)) on X n × X n define an association scheme Qua(n, q). The orbits of GL n(\(\mathbb{F}_q \)) 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).