Jordan Type stratification of spaces 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. Košir 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 $Q$.