A Spectrahedral Representation for Polar Orbitopes

Let $G$ be a Lie group with real semisimple Lie algebra $\mathfrak{g}$. Further let $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ be a Cartan decomposition. The maximal compact subgroup $K\subseteq G$ acts on $\mathfrak{p}$ via the adjoint representation and the convex hulls of the resulting orbits are the polar orbitopes. We prove that every polar orbitope is a spectrahedron by giving an explicit representation. In addition we give a new proof for the fact that the faces of a polar orbitope are, up to conjugation, given by the faces of the momentum polytope.