Subspaces fixed by a nilpotent matrix

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