Counter Example to a Strong Matroid Minor Conjecture

TL;DR

This paper provides a counterexample to a strong version of the Matroid Minor Conjecture, demonstrating that certain ascending chains of ideals exist in the affine infinite Grassmannian, challenging previous assumptions.

Contribution

It constructs explicit counterexamples to a conjecture about the structure of ideals in the affine infinite Grassmannian, disproving the conjecture's strong form.

Findings

Existence of properly ascending chains of $S_inity$-stable ideals

Counterexample to the strong Matroid Minor Conjecture

Open questions on topological noetherian property remain

Abstract

The main result of this note asserts that a strong form of the Matroid Minor Conjecture due to J. Draisma is not true, i.e., there exist properly ascending chains of $S_\infty$-stable ideals in the affine coordinate ring of the affine infinite Grassmannian, where $S_\infty$ is the infinite symmetric group. In fact, we explicitly construct such an ascending chain. His conjectures on topological noetherian property for the affine infinite Grassmannian remain open though.