Non-Abelian momentum polytopes for products of $CP ^2$

TL;DR

This paper classifies the momentum polytopes arising from the action of SU(3) on products of complex projective 4-space, revealing various types depending on symplectic form weights, as part of a study on higher-dimensional vortex systems.

Contribution

It provides a complete classification of the momentum polytopes for SU(3) actions on products of CP^4, extending understanding of symplectic geometry in higher dimensions.

Findings

Identified 8 types of generic momentum polytopes for 3 copies of CP^4.

Classified transition polytopes based on symplectic form weights.

Established inequalities for eigenvalues of sums of Hermitian matrices.

Abstract

This is the first of two companion papers. The joint aim is to study a generalization to higher dimension of the point vortex systems familiar in 2-D. In this paper we classify the momentum polytopes for the action of the Lie group SU(3) on products of copies of complex projective 4-space. For 2 copies, the momentum polytope is simply a line segment, which can sit in the positive Weyl chamber in a small number of ways. For a product of 3 copies there are 8 different types of generic momentum polytope, and numerous transition polytopes, all of which are classified here. The type of polytope depends on the weights of the symplectic form on each copy of projective space. In the second paper we use techniques of symplectic reduction to study the possible dynamics of interacting generalized point vortices. The results can be applied to determine the inequalities satisfied by the eigenvalues of the sum of up to three 3x3 Hermitian matrices with double eigenvalues.