  Let $G$ be the group $GL_r(C) \times (C^\times)^n$. We conjecture that the
finely-graded Hilbert series of a $G$ orbit closure in the space of $r$-by-$n$
matrices is wholly determined by the associated matroid. In support of this, we
prove that the coefficients of this Hilbert series corresponding to certain
hook-shaped Schur functions in the $GL_r(C)$ variables are determined by the
matroid, and that the orbit closure has a set-theoretic system of ideal
generators whose combinatorics are also so determined. We also discuss
relations between these Hilbert series for related matrices, including their
stabilizing behaviour as $r$ increases.
