Representation of $\mathfrak{sl}_4\mathbb{C}$: exercise 15.12 in Fulton & Harris V is standard representation of $\mathfrak{sl}_4\mathbb{C}$ on $\mathbb{C^4}$. $\wedge^2V$ is the irreducible representation $\Gamma_{0,1,0}$.
Exercise is to prove the kernel of $\phi_n$ will be  $\Gamma_{0,n,0}$
\begin{align}
\phi_n: \text{Sym}^n(\wedge^2V) \to \text{Sym}^{n-2}(\wedge^2V)
\end{align}
Actually if we act primitive negative root vectors on the highest weight vector $(e_1\wedge e_2)^n$ to generate the irreducible representation $\Gamma_{0,n,0}$, then we can do similar thing time and time again until we get an irreducible decomposition of $\text{Sym}^n(\wedge^2V)$. The process seems tedious and the multiplicity...
Another question is how the map actually work? I can not give an example which map $\text{Sym}^n(\wedge^2V)$ to $\text{Sym}^{n-2}(\wedge^2V)$.
 A: For me, the natural way to see this is that there is a natural symmetric bilinear form on $\bigwedge^2 V$:
$$ (v\wedge w, x\wedge y) := v\wedge w\wedge x\wedge y \in \bigwedge\nolimits^4 V \cong \mathbb{C}.$$
Indeed this idea actually identifies $\mathfrak{sl}_4$ with $\mathfrak{so}_6$.
A symmetric bilinear form is an element $ \omega \in\operatorname{Sym}^2(\bigwedge^2 V^*) \cong \operatorname{Sym}^2(\bigwedge^2 V)^*$, so $\operatorname{Sym}^n(\bigwedge^2 V)$ admits a contraction map by this form:
$\phi^n(\alpha_1 \odot \cdots \odot \alpha_n) := \omega \lrcorner (\alpha_1 \odot \cdots \odot \alpha_n) \in \operatorname{Sym}^{n-2}(\bigwedge^2 V)$.
If this is unclear look at how this is done for symmetric powers of the defining representation of $\mathfrak{so}_n$ or alternating powers of the defining representation of $\mathfrak{sp}_{2n}$. I'm sure Fulton and Harris go into detail on this at some point.
Edit: Having had a quick scan through the book the other examples I mentioned are given in Exercises 16.11 and 18.7. If you are not familiar with contraction, they describe it explicitly for the $\mathfrak{sp}_{2n}$ case around Theorem 17.5 and theres a whole section on it in the appendix.
