## Abstract

This is the first of two companion papers. The joint aim is to study

a generalization to higher dimension of the familiar point vortex systems in 2

dimensions. In this paper we classify the momentum polytopes for the action

of the Lie group SU(3) on products of copies of complex projective 2-space (a

real 4-dimensional manifold). 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 of this paper can be applied to determine the inequalities satisfied by the eigenvalues of the sum of up to three 3x3 Hermitian matrices where

each has a double eigenvalue

a generalization to higher dimension of the familiar point vortex systems in 2

dimensions. In this paper we classify the momentum polytopes for the action

of the Lie group SU(3) on products of copies of complex projective 2-space (a

real 4-dimensional manifold). 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 of this paper can be applied to determine the inequalities satisfied by the eigenvalues of the sum of up to three 3x3 Hermitian matrices where

each has a double eigenvalue

Original language | English |
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Pages (from-to) | 575 |

Number of pages | 599 |

Journal | Journal of Geometric Mechanics |

Volume | 11 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Dec 2019 |

## Keywords

- Momentum map
- eigenvalue estimates
- convex polyhedra
- symplectic geometry