The University of Michigan Combinatorics Seminar
Fall 2009
September 25, 4:10-5:00, 3866 East Hall



Orbitopes

Frank Sottile

Texas A&M University


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 low-dimensional 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 trace-free symmetric matrices.

This is joint work with Raman Sanyal and Bernd Sturmfels.