[Paper Review] Reducibility of nilpotent commuting varieties

Robert M. Guralnick, Nham V. Ngo|arXiv (Cornell University)|Aug 11, 2013

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes that the nilpotent commuting variety $C_r(\mathcal{N}_n)$ is reducible for all $n, r \geq 4$, using representation-theoretic and geometric techniques involving parabolic subgroups and orbit closures. It proves that reducibility of $C_r(\mathfrak{gl}_n)$ implies that of $C_r(\mathcal{N}_n)$ under certain conditions, and provides new lower bounds for the dimensions of both $C_r(\mathcal{N}_n)$ and $C_r(\mathfrak{gl}_n)$, significantly improving prior estimates.

ABSTRACT

Let $\N_n$ be the set of nilpotent $n$ by $n$ matrices over an algebraically closed field $k$. For each $r\ge 2$, let $C_r(\N_n)$ be the variety consisting of all pairwise commuting $r$-tuples of nilpotent matrices. It is well-kown that $C_2(\N_n)$ is irreducible for every $n$. We study in this note the reducibility of $C_r(\N_n)$ 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 $C_r(\N_n)$ under certain condition. Then we prove that $C_r(\N_n)$ 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 $C_r(\N_n)$ and $C_r(\mathfrak{gl}_n)$. Finally, we investigate values of $n$ for which the variety $C_3(\N_n)$ of nilpotent commuting triples is reducible.

Motivation & Objective

• To determine the reducibility status of nilpotent commuting varieties $C_r(\mathcal{N}_n)$ for $r \geq 3$, particularly for higher ranks.
• To investigate the relationship between the reducibility of $C_r(\mathfrak{gl}_n)$ and $C_r(\mathcal{N}_n)$, testing the conjecture that their reducibility behaviors are similar.
• To establish new lower bounds for the dimensions of $C_r(\mathcal{N}_n)$ and $C_r(\mathfrak{gl}_n)$ using orbit closures of parabolic subgroups.
• To narrow the interval for $n'_3$, the smallest $n$ such that $C_3(\mathcal{N}_n)$ is reducible, improving upon previous estimates.

Proposed method

• Using the orbit closure $G \cdot \mathfrak{u}_P^r$ for a parabolic subgroup $P$ of $GL_n(k)$, where $\mathfrak{u}_P$ is the Lie algebra of the unipotent radical of $P$.
• Computing the dimension of $V_P = G \cdot \mathfrak{u}_P^r$ via fiber dimension theorems, leading to a lower bound for $\dim C_r(\mathcal{N}_n)$.
• Applying a criterion that if $C_r(\mathcal{N}_n)$ is irreducible, then the subalgebra generated by any $r$-tuple in it has dimension at most $n-1$, to derive a contradiction for $n,r \geq 4$.
• Constructing a specific nilpotent matrix $v$ of size $4s \times 4s$ with a centralizer $z(v)$ containing a subspace $\Gamma$ of dimension $4s^2$, and analyzing $C_2(\Gamma)$ to bound $\dim C_3(\mathcal{N}_n)$.
• Using the inequality $\dim C_3(\mathcal{N}_n) \geq n^2 + s^2$ and comparing it with the known upper bound $n^2 + n - 2$ for irreducible $C_3(\mathcal{N}_n)$ to derive a contradiction when $s \geq 4$.
• Generalizing the method from $r=3$ to $r>3$ by showing that $C_r(\mathcal{N}_n)$ is reducible for all $n,r \geq 4$ using similar orbit and dimension arguments.

Experimental results

Research questions

• RQ1 Is $C_r(\mathcal{N}_n)$ reducible for all $n, r \geq 4$?
• RQ2 Does the reducibility of $C_r(\mathfrak{gl}_n)$ imply the reducibility of $C_r(\mathcal{N}_n)$ under certain conditions?
• RQ3 What is the smallest $n$ such that $C_3(\mathcal{N}_n)$ is reducible, and can this value be bounded below 30?
• RQ4 Can new lower bounds for the dimensions of $C_r(\mathcal{N}_n)$ and $C_r(\mathfrak{gl}_n)$ be established using orbit closures of parabolic subgroups?
• RQ5 Is $C_r(\mathcal{N}_n)$ equidimensional, or do multiple irreducible components of different dimensions exist?

Key findings

• The nilpotent commuting variety $C_r(\mathcal{N}_n)$ is reducible for all $n, r \geq 4$, resolving a key open case in the theory of commuting varieties.
• For $r > 3$, the reducibility of $C_r(\mathfrak{gl}_n)$ implies that of $C_r(\mathcal{N}_n)$ under suitable conditions, and $n'_r = 4$ for $r > 3$, matching $n_r$.
• The upper bound for $n'_3$, the smallest $n$ such that $C_3(\mathcal{N}_n)$ is reducible, is improved to $n'_3 \leq 16$, significantly narrowing the interval from $[10,30]$.
• A new lower bound for $\dim C_r(\mathcal{N}_n)$ is established as $(r+1)\dim \mathfrak{u}_P$, which exceeds the classical bound $n^2 - n + (r-1)(n-1)$ for most $n, r \geq 4$.
• The variety $V_P = G \cdot \mathfrak{u}_P^r$ is shown to be an irreducible component of $C_r(\mathcal{N}_n)$ for $n,r \geq 4$, supporting the conjecture that $V_P$ is a maximal irreducible component.
• For $C_r(\mathfrak{gl}_n)$, a new lower bound of $(r+1)\dim \mathfrak{u}_P + r$ is derived, which exceeds the classical bound when $n \geq 4, r \geq 9$ or $n \geq 12, r \geq 4$.