Geometric Invariant Theory and Generalized Eigenvalue Problem II

TL;DR

This paper provides a bijective parametrization of the faces of a cone related to the generalized eigenvalue problem, connecting geometric invariant theory with representation theory, and applies these results to moment polytopes.

Contribution

It introduces a new bijective parametrization of the faces of the cone generated by dominant characters, linking geometric invariant theory with eigenvalue problems.

Findings

Parametrization of cone faces in terms of dominant characters

Conditions for face containment within larger faces

Reproof of known results on moment polytopes

Abstract

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 $LR^\circ(\hat{G},G)$ generated by the pairs $(\nu,\hat{\nu})$ of strictly dominant characters such that $V_\nu$ is a submodule of $V_{\hat\nu}$. The main result of this article is a bijective parametrisation of the faces of $LR^\circ(\hat G,G)$. We also explain when such a face is contained in another one. In way, we obtain results about the faces of the Dolgachev-Hu's $G$-ample cone. We also apply our results to reprove known results about the moment polytopes.