Matrix orbit closures

Abstract

Let G be the group $$\mathrm {GL}_r(\mathbf {C}) \times (\mathbf {C}^\times )^n.$$ 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 $$\mathrm {GL}_r(\mathbf {C})$$ 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.