On the Grassmann module of symplectic dual polar spaces of rank 4 in characteristic 3

Let $V$ be the Weyl module of dimension ${ 2n \choose n}-{2n \choose n-2}$ for the symplectic group $\Sp(2n,\F)$ whose highest weight is the $n$-th fundamental dominant weight. The module $V$ affords the grassmann embedding of the symplectic dual polar space $DW(2n-1,\F)$, therefore $V$ is also called the grassmann-module for the symplectic group. We consider the smallest case for $Char(\F)$ odd for which $V$ is reducible, namely $n = 4$ and $Char(\F)=3$. In this case the unique factor $R$ of $V$ has vector dimension $1$. Here we provide a geometric description for $R$ and study some relations between $R$ and other objects associated with the grassmann embedding.