Semidefinite Relaxations for the Grassmann Orbitope
The Grassmann orbitope is the convex hull over the Grassmann variety of decomposable skew-symmetric tensors of unit length. This variety parametrizes k-dimensional linear subspaces of Rn, and it is the highest weight orbit under the k-th exterior power representation of the group SO(n). In this talk we discuss semidefinite relaxations of the Grassmann orbitope. This convex body can be approximated and represented surprisingly well by projections of sets described by linear matrix inequalities (i.e. a projection of a so-called spectrahedron). We show how these relaxations are constructed and we discuss relations to a longstanding conjecture on calibrations by Harvey and Lawson. This is an ongoing project together with Raman Sanyal (FU Berlin).