The poset of the nilpotent commutator of a nilpotent matrix

TL;DR

This paper investigates the structure of the nilpotent commutator of a nilpotent matrix, introducing a poset and a partition that generalize previous recursive processes, ultimately showing they produce the same partition.

Contribution

It generalizes prior recursive processes to define a new partition $_U(P)$, proving it coincides with the partition obtained from the nilpotent commutator.

Findings

The partition $_U(P)$ matches the Jordan partition of a generic element in the nilpotent commutator.

The paper unifies several recursive processes under a single framework.

It provides a new combinatorial description of the nilpotent commutator's structure.

Abstract

Let $B$ be an $n \times n$ nilpotent matrix with entries in an infinite field $\k$. Assume that $B$ is in Jordan canonical form with the associated Jordan block partition $P$. In this paper, we study a poset $\mathcal{D}_P$ associated to the nilpotent commutator of $B$ and a certain partition of $n$, denoted by $\lambda_U(P)$, defined in terms of the lengths of unions of special chains in $\mathcal{D}_P$. Polona Oblak associated to a given partition $P$ another partition $Ob(P)$ resulting from a recursive process. She conjectured that $Ob(P)$ is the same as the Jordan partition $Q(P)$ of a generic element of the nilpotent commutator of $B$. Roberta Basili, Anthony Iarrobino and the author later generalized the process introduced by Oblak. In this paper we show that all such processes result in the partition $\lambda_U(P)$.