  Let $\mathbb{G}(d,n)$ be the complex Grassmannian of affine $d$-planes in
$n$-space. We study the problem of characterizing the set of algebraic
subvarieties of $\mathbb{G}(d,n)$ invariant under the action of the maximal
torus $T$ and having given homology class $\lambda$. We give a complete answer
for the case where $\lambda$ 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: we calculate the series for 3-cycles in $\mathbb{G}(2,4)$ and
carry out partial calculations for $\mathbb{G}(2,5)$.
