Geometric invariant theory and the generalized eigenvalue problem

By Nicolas Ressayre. 2007

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Let G be a connected reductive subgroup of a complex connected reductive group $\hat{G}$. Fix maximal tori and Borel subgroups of G and ${\hat{G}}$. Consider the cone $\mathcal{LR}(G,{\hat{G}})$ generated by the pairs $(\nu,{\hat{\nu}})$ of dominant characters such that $V_{\nu}^{*}$ is a submodule of $V_{{\hat{\nu}}}$ (with usual notation). Here we give a minimal set of inequalities describing $\mathcal{LR}(G,{\hat{G}})$ as a part of the dominant chamber. In other words, we describe the facets of $\mathcal{LR}(G,{\hat{G}})$ which intersect the interior of the dominant chamber. We also describe smaller faces. Finally, we are interested in some classical redundant inequalities.Along the way, we obtain results about the faces of the Dolgachev-Hu G-ample cone and variations of this cone.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5bda092636360b284dddde038286ed4edc53187