The alternating-form symmetric-operator equality conjecture

At least 7 years old · documented by 2018

Let $b$ be a non-degenerate alternating bilinear form on an $n$ -dimensional vector space $V$ over a field with characteristic not $2$ . Set

\nu:=\frac{n}{2},

and let $\mathcal{V}$ be a nilpotent linear subspace of $\mathcal{S}_b$ with dimension $\nu(n-\nu)$ . A maximal partially complete $b$ -singular flag is denoted by $\mathcal{F}$ , and $\mathcal{N}\mathcal{S}_{b,\mathcal{F}}$ denotes the associated nilpotent subspace.

Alternating-form symmetric-operator equality conjecture. There exists a maximal partially complete $b$ -singular flag $\mathcal{F}$ of $V$ such that

\mathcal{V}=\mathcal{N}\mathcal{S}_{b,\mathcal{F}}.

This is one of the two equality cases not covered by the proved results because $\mathcal{S}_b$ is generally not stable under squares when $b$ is alternating; the conjecture asserts that the corresponding classification still holds.

References

Primary source

Clément de Seguins Pazzis, “The structured Gerstenhaber problem (I)”, arXiv:1804.07938 (2018).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A 2018 paper left the conjecture open but announced a proof over sufficiently large fields, while a later posted claim says that proof appeared; neither claim has been independently verified, so the general problem remains open.

The conjecture classifies maximal-dimensional nilpotent subspaces of operators symmetric for a non-degenerate alternating form: each should arise from a maximal partially complete singular flag. de Seguins Pazzis stated the conjecture in 2018 and explicitly did not prove it in the paper.

April 2018 conditional claim

The paper announces that the conjecture holds when the field has sufficiently large cardinality relative to the Witt index, with details deferred to a subsequent article. This is a conditional claim, not a proof supplied in the paper.

Posted attempt

A later posted claim says that arXiv:1806.11355 proves the conjecture under a mild field-cardinality assumption, leaving only finite fields of odd characteristic with small cardinality relative to $n$ . The claimed proof has not been independently verified here.

Current status (as of August 2026): The conjecture has an unverified claimed resolution under a large-field condition; no verified proof is recorded, and the unrestricted case remains open.

Sources

arxiv.org

Partial progressThis solution needs a summarySee full solutionHide full solution

The conjecture was solved in the subsequent paper https://arxiv.org/pdf/1806.11355 under a mild cardinality assumption on the underlying field (with characteristic other than $2$ ). The question remains open only for finite fields with odd characteristic and small cardinality with respect to $n$ .