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?
2026-03-25 16:09:38.1774454978
How can I pick out the symmetric and antisymmetric parts of a tensor product of line bundles over projective space?
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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.