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When one is given a partition $\lambda=(\lambda_1,...,\lambda_r)$ and a locally free sheaf $\mathcal{E}$ on for example a Grassmannian variety one can apply the Schur-functor $\Sigma^{\lambda}(\mathcal{E})$ for some partition $\lambda$. Now take an invertible sheaf $\mathcal{L}$ and my question is: What is $\Sigma^{\lambda}(\mathcal{E}\otimes \mathcal{L})$ in terms of $\Sigma^{\lambda}(\mathcal{E})$ and $\mathcal{L}$ or $\Sigma^{\lambda}(\mathcal{L})$? Is ther a formula for computing Schur-functor of tensorproduct?

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There is a formula. Let $ M $ be a square matrix, $ \lambda $ a partition of $n$. We can think of $M$ as a map of free modules, so it makes sense to think about $S_{\lambda} M$. The entries of $ S_{\lambda} M $ are homogeneous polynomials in the entries of $ M $ with degree $ n$. If $M$ is a transition matrix for the vector bundle $E$, then to get the transition matrix $E \otimes L $ you multiply every entry of $M$ by the transition matrix for $L$. Therefore to get the transition matrix of $S_{\lambda} (E \otimes L) $ you take the transition matrix for $S_{\lambda} E $ and multiply each entry by the transition matrix of $L^{\otimes n}$. This tells us that $ S_{\lambda}(E \otimes L) = (S_{\lambda} E) \otimes L^{\otimes n}$

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