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

About 13 years old · traced to 2013

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 1Other wordings

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).

References

Primary source

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

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Monotonicity conjecture for reducibility of nilpotent commuting varieties

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