Matroids and Geometric Invariant Theory of torus actions on flag spaces

Benjamin J. Howard

Published 2005-11-24, updated 2006-02-14Version 2

We apply a theorem of Gel'fand, Goresky, MacPherson, and Serganova about matroid polytopes to study semistability of partial flags relative to a T-linearized ample line bundle of a flag space F = SL(n)/P where T is a maximal torus in SL(n) and P is a parabolic subgroup containing T. We find that the semistable points are all detected by invariant sections of degree one regardless of the line bundle or linearization thereof, provided there exists at least one nonzero invariant section of degree one. In this case the degree one sections are sufficient to give a well defined map from the G.I.T. quotient F//T to projective space. Additionally, we show that the closure of any T-orbit in SL(n)/P is a projectively normal toric variety for any projective embedding of SL(n)/P.