Non-abelian momentum polytopes for products of CP2

James Montaldi, Amna Shaddad

Research output: Contribution to journalArticlepeer-review


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
Original languageEnglish
Pages (from-to)575
Number of pages599
JournalJournal of Geometric Mechanics
Issue number4
Publication statusPublished - 1 Dec 2019


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


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