It is well-known that $\mathfrak{sl}_2(\mathbb{C})$ admits exactly one irreducible representation $V_n$ of dimension $n+1$ for all $n\geq 0$. It is explicitly given by the action on homogeneous polynomials in two variables of degree $n$. We will concentrate on even $n$ here.

Also well-known is the Clebsch-Gordan decomposition: for $m\leq n$ we have $V_{2n}\otimes V_{2m} \cong \bigoplus_{k=0}^{n-m} V_{2n+2m-2k}.$ A similar formula holds for the anti-symmetric part of the tensor product, the exterior product: $$V_{2n}\wedge V_{2m}\cong \bigoplus_{n+m+k \text{ odd}} V_{2n+2m-2k}.$$

My question is: *in terms of homogeneous polynomials, can one write down the projection map from $V_{2n}\wedge V_{2m}$ to one of the factors?*

In the book of Fulton-Harris on representation theory, in Exercise 11.29 and 11.30, the claim is that the projection of $V_{2n}\wedge V_{2n} = \Lambda^2(V_{2n})$ onto $V_{4n-2}$ is given by $F\wedge G \mapsto F\cdot dG-G\cdot dF$. This can't be true since $F\cdot dG-G\cdot dF$ is a polynomial of degree $4n-1$, not $4n-2$. I think one has to divide by $x dy-y dx$. Is that true? Is there a similar formula for the projection on the other factors?

Hitchin in his paper "Lie groups and Teichmüller space", in Section 6, writes that the projection in the Clebsch-Gordan decomposition is given by "contracting $k$ times with the symplectic form". What does this mean?