Double Covers of Symplectic Dual Polar Graphs

Abstract

Let $\Gamma =\Gamma (2n,q)$ be the dual polar graph of type $Sp(2n,q)$. Underlying this graph is a $2n$-dimensional vector space $V$ over a field ${\mathbb{F}}_q$ of odd order $q$, together with a symplectic (i.e. nondegenerate alternating bilinear) form $B:V\times V\to {\mathbb{F}}_q$. The vertex set of $\Gamma$ is the set $\mathcal{V}$ of all $n$-dimensional totally isotropic subspaces of $V$. If $q\equiv 1$ mod 4, we obtain from $\Gamma$ a nontrivial two-graph $\Delta =\Delta (2n,q)$ on $\mathcal{V}$ invariant under $PSp(2n,q)$. This two-graph corresponds to a double cover $\hat{\Gamma }\to \Gamma$ on which is naturally defined a $Q$-polynomial $(2n+1)$-class association scheme on $2|\mathcal{V}|$ vertices.