Polar orbitopes

TL;DR

This paper investigates the geometric and face structure of polar orbitopes, revealing that all faces are exposed, correspond to momentum polytopes, and relate to parabolic subgroups, advancing understanding of their convex geometry.

Contribution

It provides a detailed analysis of the face structure of polar orbitopes, showing all faces are exposed and linked to momentum polytopes and parabolic subgroup orbits.

Findings

All faces are exposed and are themselves polar orbitopes.

Faces are determined by the momentum polytope.

Extreme points of faces correspond to closed orbits of parabolic subgroups.

Abstract

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum polytope. There is a tight relation with parabolic subgroups: the set of extreme points of a face is the closed orbit of a parabolic subgroup of G and for any parabolic subgroup the closed orbit is of this form.