The University of Michigan Combinatorics Seminar


Abstract 

An orbitope is the convex hull of an orbit of a compact group G acting linearly on a vector space. Orbitopes are the simplest convex bodies which possess many symmetries. Some, particularly those in lowdimensional representations of G, have very beautiful structure. Our interest is in whether or not these appealing convex bodies are spectahedra, that is, if they are described by a system of linear matrix inequalities, preferably with coefficients in the field of definition of the orbitope. In this talk, I will introduce orbitopes and discuss spectahedra and the new field of convex algebraic geometry in which these questions lie. I will illustrate this with orbitopes for SO(2) and for the special orthogonal group acting on tracefree symmetric matrices. This is joint work with Raman Sanyal and Bernd Sturmfels. 