A Spectrahedral Representation for Polar Orbitopes

TL;DR

This paper proves that all polar orbitopes, convex hulls of orbits under a Lie group action, can be represented as spectrahedra, providing explicit forms and new insights into their facial structure.

Contribution

It establishes that every polar orbitope admits a spectrahedral representation and offers a new proof relating their faces to momentum polytopes.

Findings

Every polar orbitope is a spectrahedron.

Explicit spectrahedral representations are provided.

Faces of polar orbitopes correspond to faces of momentum polytopes.

Abstract

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.