Spectrahedral and semidefinite representability of orbitopes

Abstract

Polar orbitopes are a rich class of orbitopes such as the symmetric and skewsymmetric Schur-Horn orbitopes, the Fan orbitopes and the tautological orbitope of the special orthogonal group. Our main result in Chapter 3.1 is to show that every polar orbitope under a connected group is a spectrahedron (Theorem 3.1.19). It follows that the polar orbitopes under connected groups are basic closed semialgebraic und their faces are exposed. Another consequence is that every polar orbitope is a spectrahedral shadow (Theorem 3.1.26). Our main result for Chapter 4 is Theorem 4.3.4. The theorem generalizes the fact, that every orbitope under the torus is a spectrahedral shadow and gives new examples of orbitopes under the bitorus, which are spectrahedra. Chapter 5.1 is concerned with finding orbitopes, which are not spectrahedral shadows. Our main result here, is to prove that many 30-dimensional orbitopes under the bitorus are not spectrahedral shadows (Theorem 5.2.2).