Torus Orbits in Full Flag Varieties

Matroids are central players in modern combinatorial algebraic geometry. One algebro-geometric way to find a matroid is by taking any point in the Grassmannian. Then there is an associated matroid that corresponds to the set of Plücker coordinates of that point which are nonzero. Furthermore, for a certain canonical torus action, the moment map of the orbit of any point in the Grassmannian yields a polytope associated with the matroid of the point called the matroid base polytope. We will discuss a similar story for orbits of points in full flag varieties (i.e. flag varieties for flags of subspaces of dimension 0, 1, 2, ..., k of a space of dimension n). Then I will describe my recent work investigating the combinatorics of their moment polytopes and showing that the corresponding toric varieties are smooth. Based on joint work with Raman Sanyal.