Spectrahedral representation of polar orbitopes

TL;DR

This paper demonstrates that polar orbitopes arising from polar representations of compact Lie groups are spectrahedra, providing explicit matrix inequalities and exploring their duals, which leads to new classes of doubly spectrahedral convex sets.

Contribution

It establishes that polar orbitopes are spectrahedra with explicit representations and characterizes their duals, expanding the understanding of convexity in Lie group actions.

Findings

Polar orbitopes are spectrahedra with explicit LMIs.

The duals of polar orbitopes are convex hulls of finitely many orbits.

Many dual orbitopes are also spectrahedra, creating new doubly spectrahedral families.

Abstract

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