Subspaces Fixed by a Nilpotent Matrix

TL;DR

This paper classifies subspaces fixed by nilpotent matrices for small sizes, proves a conjecture about their defining ideals up to size 7, and shows the conjecture fails at size 8, leaving open questions for affine cases.

Contribution

It confirms the shuffle equations conjecture for nilpotent matrices of size up to 7 and disproves it at size 8, advancing understanding of fixed subspace varieties.

Findings

Confirmed the conjecture for n ≤ 7

Disproved the conjecture at n=8

Open questions remain for affine Grassmannian cases

Abstract

The linear spaces that are fixed by a given nilpotent $n \times n$ matrix form a subvariety of the Grassmannian. We classify these varieties for small $n$. Mutiah, Weekes and Yacobi conjectured that their radical ideals are generated by certain linear forms known as shuffle equations. We prove this conjecture for $n \leq 7$, and we disprove it for $n=8$. The question remains open for nilpotent matrices arising from the affine Grassmannian.