Polar orbitopes

We study \textit{polar orbitopes}, i.e., convex hulls of orbits of a polar\break representation of a compact Lie group. They are given by representations of $K$ on $\mathfrak{p}$, where $K$ is a maximal compact subgroup of a real semisimple Lie group $G$ with Lie algebra $ \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$. 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.

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Published 5 July 2013