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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 algebraicsubvarieties of $\mathbb{G}(d,n)$ invariant under the action of the maximaltorus $T$ and having given homology class $\lambda$. We give a complete answerfor the case where $\lambda$ is the class of a $T$-orbit, and partial resultsfor other cases, using techniques inspired by matroid theory. This problem hasapplications to the computation of the Euler-Chow series for Grassmannians ofprojective lines: we calculate the series for 3-cycles in $\mathbb{G}(2,4)$ andcarry out partial calculations for $\mathbb{G}(2,5)$.

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