In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, the sequence $$ 0 \rightarrow \bigwedge^2 \Omega(1) \rightarrow \Omega(1) \otimes \Omega(1) \rightarrow S^2 \Omega(1) \rightarrow 0, $$ and the software Shur which does a good job of computing tensor products and plethysms. If I remember right I got that it was the irreducible representation $S^{4,2}(W)$, where $W = H^0(O_{\mathbb P^2}(1))$ is the standard representation and $S^{4,2}$ is the Shur functor.
Now I want to make a more general calculation of things of the form $H^0(S^a(S^2\Omega(1))\otimes \mathcal O(d))$. (These correspond to sections of line bundles on the Hilbert scheme of two points, which can be viewed as $\mathbb P(S^2\Omega(1))$). Shur handles compositions of symmetric products fine, but I think that in general sections of symmetric products of vector bundles are not the same as symmetric products of sections.
QUESTION What strategy could I use to determine what representation something like this is? It would be great to have a formula in terms of $a$ and $d$, but also algorithms to compute specific examples would still be interesting.
