Double covers of symplectic dual polar graphs

Abstract

Let Γ=Γ(2n,q) be the dual polar graph of type Sp(2n,q). Underlying this graph is a 2n-dimensional vector space V over a field Fq of odd order q, together with a symplectic (i.e. nondegenerate alternating bilinear) form B:V×V→Fq. The vertex set of Γ is the set V of all n-dimensional totally isotropic subspaces of V. If q≡1 mod 4, we obtain from Γ a nontrivial two-graph Δ=Δ(2n,q) on V invariant under PSp(2n,q). This two-graph corresponds to a double cover Γ̂→Γ on which is naturally defined a Q-polynomial (2n+1)-class association scheme on 2|V̂| vertices.