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 p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g=k⊕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.
Biliotti, L., Ghigi, A., Heinzner, P. (2013). Polar orbitopes. COMMUNICATIONS IN ANALYSIS AND GEOMETRY, 21(3), 579-606 [10.4310/CAG.2013.v21.n3.a5].
Polar orbitopes
GHIGI, ALESSANDRO CALLISTO;
2013
Abstract
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 p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g=k⊕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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