Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class

Let $G(d,n)$ be the complex Grassmannian of affine d-planes in n-space. We study the problem of characterizing the set of algebraic subvarieties of $G(d,n)$ invariant under the action of the maximal torus T and having given homology class λ. We give a complete answer for the case where λ is the class of a T-orbit, and partial results for other cases, using techniques inspired by matroid theory. This problem has applications to the computation of the Euler-Chow series for Grassmannians of projective lines.