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Abstract: Let be a Geometric Invariant Theory quotient of a partial flag variety by the action of the maximal torus T in , where P is a parabolic subgroup containing T. The construction of depends upon the choice of a T-linearized line bundle L of F. This note concerns the case 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 denotes the irreducible representation of with highest weight λ, and is the isotypic component corresponding to a weight μ of the torus, then is equal to . The weight μ is used to twist the canonical T-linearization of , where the canonical T-linearization of is obtained by restricting the unique -linearization of to T. We apply a theorem of Gel''fand, Goresky, MacPherson, and Serganova concerning matroid polytopes to show that if then one gets a well-defined map by taking any basis of . Equivalently, all the semistable partial flags are detected by degree one T-invariants provided is nonzero. We also show that the closure of any T-orbit in F is projectively normal for the projective embedding . [Copyright &y& Elsevier]