辛对偶极图的双覆盖
Double Covers of Symplectic Dual Polar Graphs
摘要：
设$Γ=Γ(2n,q)$是$Sp(2n,q)$型的对偶极图。在这个图的下面是一个$2n$维向量空间$V$在一个域${\mathbb F}_q$的奇数阶$q$，连同一个辛（即非退化交替双线性）形式$B:V\times V\to{\mathbb F}_q$。$Γ$的顶点集是$V$的所有$n$维全迷向子空间的集合${\mathcal V}$.如果$q\equiv1$ mod 4，则从$Γ$得到一个非平凡的二图$Δ=Δ(2n,q)$，它是关于${\mathcal V}$在$PSp(2n,q)$下不变的.这两个图对应于一个双覆盖$\widehatΓ\toΓ$上自然地定义了一个$Q$多项式$(2n+1)$类关联方案的$2|{\mathcal V}|$顶点。
英文摘要：
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 ${\mathbb F}_q$ 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 $Γ$ is the set ${\mathcal V}$ of all $n$-dimensional totally isotropic subspaces of $V$. If $q\equiv1$ mod 4, we obtain from $Γ$ a nontrivial two-graph $Δ=Δ(2n,q)$ on ${\mathcal V}$ invariant under $PSp(2n,q)$. This two-graph corresponds to a double cover $\widehatΓ\toΓ$ on which is naturally defined a $Q$-polynomial $(2n+1)$-class association scheme on $2|{\mathcal V}|$ vertices.
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