Orbitopes

Frank Sottile

  An orbitope is the convex hull of an orbit of a compact
group G acting linearly on a vector space V.   Orbitopes are 
the simplest convex bodies which possess many symmetries.
Some, particularly when V is a small representation of 
have very beautiful structure, while for others it is
hopeless to understand their structure (think the permutahedron
vs. the Birkhoff polytope).  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 the special orthogonal group acting
on trace-free symmetric matrices, for which we understand
everything, and mention a concrete open question for SO(3).
This is joint work with Raman Sanyal and Bernd Sturmfels.