Spectrahedral representation of polar orbitopes

Abstract

Let K be a compact Lie group and V a finite-dimensional representation of K. The orbitope of a vector x∈V is the convex hull Ox of the orbit Kx in V. We show that if V is polar then Ox is a spectrahedron, and we produce an explicit linear matrix inequality representation. We also consider the coorbitope Oox, which is the convex set polar to Ox. We prove that Oox is the convex hull of finitely many K-orbits, and we identify the cases in which Oox is itself an orbitope. In these cases one has Oox=c⋅Ox with c>0. Moreover we show that if x has “rational coefficients” then Oox is again a spectrahedron. This provides many new families of doubly spectrahedral orbitopes. All polar orbitopes that are derived from classical semisimple Lie algebras can be described in terms of conditions on singular values and Ky Fan matrix norms.