Reducibility of nilpotent commuting varieties

Abstract

Let ${\mathbb{N}}_n$ be the set of nilpotent $n$ by $n$ matrices over an algebraically closed field $k$. For each $r\ge 2$, let $C_r({\mathbb{N}}_n)$ be the variety consisting of all pairwise commuting $r$-tuples of nilpotent matrices. It is well-kown that $C_2({\mathbb{N}}_n)$ is irreducible for every $n$. We study in this note the reducibility of $C_r({\mathbb{N}}_n)$ for various values of $n$ and $r$. In particular it will be shown that the reducibility of $C_r({\mathfrak{gl}}_n)$, the variety of commuting $r$-tuples of $n$ by $n$ matrices, implies that of $C_r({\mathbb{N}}_n)$ under certain condition. Then we prove that $C_r({\mathbb{N}}_n)$ is reducible for all $n,r\ge 4$. The ingredients of this result are also useful for getting a new lower bound of the dimensions of $C_r({\mathbb{N}}_n)$ and $C_r({\mathfrak{gl}}_n)$. Finally, we investigate values of $n$ for which the variety $C_3({\mathbb{N}}_n)$ of nilpotent commuting triples is reducible.

8 pages

References in corpus (1)

• Commuting varieties of $r$-tuples over Lie algebras

Cited by in corpus (1)

• On varieties of commuting nilpotent matrices