On varieties of commuting nilpotent matrices

TL;DR

This paper investigates the geometric structure of varieties of commuting nilpotent matrices, establishing conditions for their irreducibility or reducibility across different dimensions and matrix sizes.

Contribution

It proves that the variety of commuting nilpotent matrices is reducible for all dimensions and sizes greater than or equal to 4, and identifies cases of irreducibility for small sizes.

Findings

N(d,n) is reducible for all d,n ≥ 4

N(3,n) is irreducible for n ≤ 6

N(d,n) is irreducible for small sizes, reducible for larger ones

Abstract

Let $N(d,n)$ be the variety of all $d$-tuples of commuting nilpotent $n\times n$ matrices. It is well-known that $N(d,n)$ is irreducible if $d=2$, if $n\le 3$ or if $d=3$ and $n=4$. On the other hand $N(3,n)$ is known to be reducible for $n\ge 13$. We study in this paper the reducibility of $N(d,n)$ for various values of $d$ and $n$. In particular, we prove that $N(d,n)$ is reducible for all $d,n\ge 4$. In the case $d=3$, we show that it is irreducible for $n\le 6$.