Counter Example to a Strong Matroid Minor Conjecture
Shrawan Kumar

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 -stable ideals in the affine coordinate ring of the affine infinite Grassmannian, where 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.
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Taxonomy
TopicsMatrix Theory and Algorithms · Advanced Algebra and Logic · Advanced Topics in Algebra
