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For the Grassmannians Kapranov proved that there is a tilting bundle consisting of $\sum^{\lambda}(\mathcal{T})$, where $\mathcal{T}$ is the tautological bundle and $\sum^{\lambda}$ the Schur functor. I want to learn somthing more on Schur functors, especially for sheaves. Does anybody know a good reference ?

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    $\begingroup$ Fulton-Harris Representation Theory is excellent. Once you understand it for vector spaces, it is just trivial to extend it to vector bundles. $\endgroup$
    – abx
    Feb 7, 2014 at 8:57

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