Let $(k,0,...,0)$ denote the highest weights vector of an irreducible representation of $\text{Sp}(2n,\mathbb{C})$. I read in Fulton-Harris, that this representation may be obtained as a direct summand of $\text{Sym}^kV$, where $V$ is the fundamental representation of $\text{Sp}(2n,\mathbb{C})$.

Is there a simple way to describe $(k,0,...,0)$ (as a subrepresentation of $\text{Sym}^kV$) as the kernel of some explicit map ($\text{Sym}^kV\longrightarrow *$)? I read that the $(0,...,0,1,0,...,0)$ representations could be constructed from the $k$'th alternating product of $V$ using the contraction maps associated to the symplectic form. I am particularly interested to know if there are known constructions of an infinite, symplectic family of irreducible representations of $\text{Sp}(2n,\mathbb{C})$ (direct summands of $V^{\otimes k}$ for $k$ odd), which project as a vector space to $\text{Sym}^{k'}V$ for some arbitrarily large $k' < k$.

A reference would also be appreciated.