Jordan Type stratification of spaces of commuting nilpotent matrices

TL;DR

This paper studies the structure of spaces of commuting nilpotent matrices, focusing on the Jordan types of their maximum nilpotent commutators, and provides explicit equations for certain loci when the stable partition has two parts.

Contribution

It determines equations defining loci of partitions in the inverse image of stable partitions with two parts, extending previous results and proposing a general conjecture.

Findings

Equations form a complete intersection for stable partitions with two parts.

The study confirms the box conjecture in specific cases.

Provides a framework for understanding the geometry of commuting nilpotent matrices.

Abstract

An $n\times n$ nilpotent matrix $B$ is determined up to conjugacy by a partition $P_B$ of $n$, its Jordan type given by the sizes of its Jordan blocks. The Jordan type $\mathfrak D(P)$ of a nilpotent matrix in the dense orbit of the nilpotent commutator of a given nilpotent matrix of Jordan type $P$ is stable - has parts differing pairwise by at least two - and was determined by R. Basili. The second two authors, with B. Van Steirteghem and R. Zhao determined a rectangular table of partitions $\mathfrak D^{-1}(Q)$ having a given stable partition $Q$ as the Jordan type of its maximum nilpotent commutator. They proposed a box conjecture, that would generalize the answer to stable partitions $Q$ having $\ell$ parts: it was proven recently by J.~Irving, T. Ko\v{s}ir and M. Mastnak. Using this result and also some tropical calculations, the authors here determine equations defining the loci of each partition in $\mathfrak D^{-1}(Q)$, when $Q$ is stable with two parts. The equations for each locus form a complete intersection. The authors propose a conjecture generalizing their result to arbitrary stable $Q$.