Matrix orbit closures

TL;DR

This paper explores how the Hilbert series of matrix orbit closures under a group action are determined by the associated matroid, providing partial proofs and discussing their algebraic and combinatorial properties.

Contribution

It proves that certain coefficients of the Hilbert series are determined by the matroid and shows the orbit closure's ideal generators are also matroid-dependent.

Findings

Coefficients of Hilbert series linked to matroids.

Orbit closure ideals have matroid-determined combinatorics.

Hilbert series exhibit stabilizing behavior as matrix size increases.

Abstract

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.