Subspaces Fixed by a Nilpotent Matrix

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 $n\le 7$, and we disprove it for $n=8$. The question remains open for nilpotent matrices arising from the affine Grassmannian.

14 pages, some updates based on referee comments