On natural representations of the symplectic group

Let $V_k$ be the Weyl module of dimension ${2n\choose k}-{2n\choose k-2}$ for the group $G = \mathrm{Sp}(2n,\mathbb{F})$ arising from the $k$-th fundamental weight of the Lie algebra of $G$. Thus, $V_k$ affords the grassmann embedding of the $k$-th symplectic polar grassmannian of the building associated to $G$. When $\mathrm{char}(\mathbb{F}) = p > 0$ and $n$ is sufficiently large compared with the difference $n-k$, the $G$-module $V_k$ is reducible. In this paper we are mainly interested in the first appearance of reducibility for a given $h := n-k$. It is known that, for given $h$ and $p$, there exists an integer $n(h,p)$ such that $V_k$ is reducible if and only if $n \geq n(h,p)$. Moreover, let $n \geq n(h,p)$ and $R_k$ the largest proper non-trivial submodule of $V_k$. Then $\mathrm{dim}(R_k) = 1$ if $n = n(h,p)$ while $\mathrm{dim}(R_k) > 1$ if $n > n(h,p)$. In this paper we will show how this result can be obtained by an investigation of a certain chain of $G$-submodules of the exterior power $W_k := \wedge^kV$, where $V = V(2n,\mathbb{F})$