Counter Example to a Strong Matroid Minor Conjecture

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 $S_{\mathrm{\infty }}$-stable ideals in the affine coordinate ring of the affine infinite Grassmannian, where $S_{\mathrm{\infty }}$ 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.