Exterior products of irreducible representations of sl_2(C)

Question

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?

Answer 1

The explicit projection of $V_m\otimes V_n$ on $V_{m+n-2k}$ is given by the $k$-th transvectant. $F\otimes G\mapsto (F,G)_k$, which with classical mid 19-th century normalization is given by $$ (F,G)_k=\frac{(m-k)!\ (n-k)!}{m!\ n!} \left.\left(\frac{\partial^2}{\partial x_1\partial y_2}-\frac{\partial^2}{\partial x_2\partial y_1}\right)^k F(x_1,x_2)G(y_1,y_2)\right|_{y:=x} $$ $$ =\frac{(m-k)!\ (n-k)!}{m!\ n!}\sum_{j=0}^{k}\binom{k}{j}(-1)^j\ \partial_1^{k-j}\partial_2^{j}F\ \partial_1^{j}\partial_2^{k-j}G $$ where $\partial_1$ denotes $\frac{\partial}{\partial x_1}$ and $\partial_2$ denotes $\frac{\partial}{\partial x_2}$.

As for the construction mentioned by Hitchin, it's just how you construct the symmetric tensor (meaning just an array of numbers) for the transvectant using the symmetric tensors of $F$ and $G$. You just contract indices with the intermediation of $k$ epsilon Levi-Civita tensors which here are matrices $$ \varepsilon=\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} $$ This is best explained in pictures. See Section 2 of my article

• A. Abdesselam, On the volume conjecture for classical spin networks. J. Knot Theory Ramifications 21 (2012), no. 3, 1250022, 62 pp.

and Section 4.2 of

• A. Abdesselam and J. Chipalkatti, On the Reconstruction Problem for Pascal Lines. Discrete & Computational Geometry 60 (2018), 381-405.

See in particular equations (4.4) and (4.5) of the last reference (the published version, not the arXiv preprint), shown in the snippet below.