Let k be an algebraically closed field, and let N_n be the variety of nilpotent n× n matrices. For r≥ 2, let C_r(N_n) be the variety of pairwise commuting r-tuples of elements of N_n. Monotonicity conjecture. If C_r(N_n) is reducible, then C_r(N_n+1) is reducible. If true, this would propagate known reducibility in matrix size n to every larger matrix size. The source presents the assertion under “Open problems” and gives no proof.

The monotonicity conjecture for reducibility of nilpotent commuting varieties

Let $k$ be an algebraically closed field, and let $\mathcal{N}_n$ be the variety of nilpotent $n\times n$ matrices. For $r\ge 2$ , let $C_r(\mathcal{N}_n)$ be the variety of pairwise commuting $r$ -tuples of elements of $\mathcal{N}_n$ .

Monotonicity conjecture. If $C_r(\mathcal{N}_n)$ is reducible, then $C_r(\mathcal{N}_{n+1})$ is reducible.

If true, this would propagate known reducibility in matrix size $n$ to every larger matrix size. The source presents the assertion under “Open problems” and gives no proof.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Monotonicity conjecture for reducibility of nilpotent commuting varieties
Let $\mathcal{N}_n$ be the variety of nilpotent $n\times n$ matrices, and let $C_r(\mathcal{N}_n)$ denote the variety of commuting $r$ -tuples in $\mathcal{N}_n$ . Monotonicity conjecture. If $C_r(\mathcal{N}_n)$ is reducible, then so is $C_r(\mathcal{N}_{n+1})$ . This would propagate known reducibility results from one matrix size to all larger sizes; the supplied text gives no resolution of the conjecture.
source: Robert M. Guralnick and Nham V. Ngo, “Reducibility of nilpotent commuting varieties”, arXiv:1308.2420 (2013).

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Robert M. Guralnick and Nham V. Ngo, “Reducibility of nilpotent commuting varieties”, arXiv:1308.2420 (2013).

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The monotonicity conjecture for reducibility of nilpotent commuting varieties

Equivalent formulations 1

Monotonicity conjecture for reducibility of nilpotent commuting varieties

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