On pairs of commuting nilpotent matrices

TL;DR

This paper investigates the structure of pairs of commuting nilpotent matrices, proving a key idempotent property of a certain map between their Jordan forms and linking generic pairs to Gorenstein algebras.

Contribution

It establishes that the map from a nilpotent matrix's partition to a unique dense orbit partition is idempotent, answering open questions and characterizing generic pairs as Gorenstein algebra generators.

Findings

The map D is idempotent.

Generic pairs generate Gorenstein algebras.

Explicit description of D for partitions with at most two parts.

Abstract

Let $B$ be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition $\lambda$. Then it is known that its nilpotent commutator $N_B$ is an irreducible variety and that there is a unique partition $\mu$ such that the intersection of the orbit of nilpotent matrices corresponding to $\mu$ with $N_B$ is dense in $N_B$. We prove that map $D$ given by $D(\lambda)=\mu$ is an idempotent map. This answers a question of Basili and Iarrobino and gives a partial answer to a question of Panyushev. In the proof, we use the fact that for a generic matrix $A \in N_B$ the algebra generated by $A$ and $B$ is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe $D(\lambda)$ in terms of $\lambda$ if $D(\lambda)$ has at most two parts.