Matrix orbit closures

Abstract

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

The proof of the main theorem (Theorem 5.1) of v5, asserting the invariance of the equivariant K-class of a matrix orbit closure, was incorrect and has been removed