Orbitopes

Frank Sottile

  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.