Jordan Type stratification of spaces of commuting nilpotent matrices

TL;DR

This work investigates the Jordan-type loci within the nilpotent commuting variety for a fixed nilpotent Jordan matrix $J_Q$, focusing on stable partitions $Q$ with two parts. By combining Burge code combinatorics and tropical linear-algebra techniques, the authors construct explicit defining equations $E^Q_{k,l}$ for the loci $W^Q_P$ (where $P$ ranges over $\mathfrak D^{-1}(Q)$) and prove that, in the two-part case $Q=(u,u-r)$ with $r\ge2$, the closures $\overline{W^Q_P}$ are irreducible complete intersections of codimension $\ell(P)-2$. The approach leverages the Box Theorem to organize $\mathfrak D^{-1}(Q)$ into a rectangular table and ties partitions to Burge codes, with tropical calculations controlling ranks and coranks of powers of commuting nilpotent matrices. The authors also formulate a general conjecture extending these results to arbitrary stable partitions $Q$, outlining an inductive path and highlighting open challenges for length $\ell\ge3$. Overall, the paper provides a concrete, complete-intersection description of two-part stable cases and advances the program to describe Jordan-type loci for commuting nilpotent matrices in broader settings.

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 $Q$.

Figures (5)

• Figure 1: Finding $\partial P$, $\partial^2 P$, and $\partial^3 P$ for $P=(7,6,5^3,2^2,1)$. Each partition is decomposed into disjoint almost rectangular subpartitions, starting from the top. In the figure, these almost rectangular subpartitions are separated from each other by thick borders within each partition. The map $\partial$ acts on a partition by removing one box from the lowest row of the largest part of each almost rectangular subpartition. Those boxes are shaded in gray for each partition.
• Figure 2: Illustration for computation of the Burge code in Proposition \ref{['prop:generic']} when $2k\leq r$.
• Figure 3: Illustration for the computation of the Burge code in Proposition \ref{['prop:generic']} when $2k>r$ and $l'=r-k$.
• Figure 4: Illustration for computation of the Burge code in Proposition \ref{['prop:generic']}
• Figure 5: Loci $W^Q_P$, for $P=P_{k,l}\in\mathcal{T}(Q), Q=(5,2)$. (Example \ref{['52ex']}).

Theorems & Definitions (29)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Example 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Remark 2.5
• Proposition 2.6
• proof