How can I pick out the symmetric and antisymmetric parts of a tensor product of line bundles over projective space? If I have a vector bundle constructed as a direct sum of line bundles
$$
E = \mathcal{O}(a_1)\oplus \cdots \oplus \mathcal{O}(a_n)
$$
Is there any general method for picking out the symmetric and antisymmetric parts of a tensor bundle of $E$? For example, how can I find the decomposition of
\begin{align*}
\otimes^3(\mathcal{O}(a)\oplus\mathcal{O}(b)) &= \mathcal{O}(3a) \oplus \mathcal{O}(2a+b)^{\oplus 2} \oplus \mathcal{O}(a+2b)^{\oplus 2}\oplus\mathcal{O}(3b) \\
&= \text{Sym}^3(E) \oplus \Lambda^3 E
\end{align*}
in terms of line bundles?

Use formulas 
$$
S^n(E_1 \oplus E_2) = \bigoplus_{i=0}^n S^i(E_1) \otimes S^{n-i}(E_2),
\qquad
\Lambda^n(E_1 \oplus E_2) = \bigoplus_{i=0}^n \Lambda^i(E_1) \otimes \Lambda^{n-i}(E_2),
$$
and iterate them.
