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Quivers and the cohomology of homogeneous vector bundles

Giorgio Ottaviani·Elena Rubei
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Abstract

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE type as the cohomology of a complex explicitly described. The main tool is the equivalence between the category of homogeneous bundles and the category of representations of a certain quiver ${\cal Q}_X$ with relations, whose vertices are the dominant weights of the reductive part of $P$. This equivalence was found in some cases by Bondal, Kapranov and Hille and we find the appropriate relations for any Hermitian symmetric variety, computing them explicitly for Grassmannians.

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Giorgio Ottaviani, Elena Rubei. 2005-04-22. Quivers and the cohomology of homogeneous vector bundles. https://arxiv.org/abs/math/0403307

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