On varieties of commuting nilpotent matrices

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 $n\le 6$.

This paper is a revision of the preprints http://arxiv.org/abs/1308.2420 (Guralnick and Ngo) and version 1 of http://arxiv.org/abs/1308.4438 (Šivic) which contain errors: 1)Proof of Thm. 3.4.1 in the first one is not completed. 2)Thm. 22 in the second one is incorrect. We fix these mistakes in this final version, to appear in Linear Algebra Appl

References in corpus (1)

• Reducibility of nilpotent commuting varieties