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 $D\left(P\right)$ 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 $D^{-1}\left(Q\right)$ 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 ℓ 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 $D^{-1}\left(Q\right)$, 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.