Matroids and Geometric Invariant Theory of torus actions on flag spaces

Let F//T be a Geometric Invariant Theory quotient of a partial flag variety by the action t gP=tgP of the maximal torus T in , where P is a parabolic subgroup containing T. The construction of F//T depends upon the choice of a T-linearized line bundle L of F. This note concerns the case L=Lλ is a very ample homogeneous line bundle determined by a dominant weight λ, meaning the associated character extends to P and to no larger parabolic subgroup. If Vλ denotes the irreducible representation of with highest weight λ, and Vλ[μ] is the isotypic component corresponding to a weight μ of the torus, then F//T is equal to . The weight μ is used to twist the canonical T-linearization of Lλ, where the canonical T-linearization of Lλ is obtained by restricting the unique -linearization of Lλ to T. We apply a theorem of Gelʹfand, Goresky, MacPherson, and Serganova concerning matroid polytopes to show that if Vλ[μ]≠0 then one gets a well-defined map by taking any basis of Vλ[μ]. Equivalently, all the semistable partial flags are detected by degree one T-invariants provided Vλ[μ] is nonzero. We also show that the closure of any T-orbit in F is projectively normal for the projective embedding .