# P2748: Largest commuting nilpotent subspaces of M5(F2)

- ID: `P2748`
- Reference: `commuting-nilpotent-subspace-m5-f2`
- Page: https://theoremdb.org/statements/P2748
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/commuting-nilpotent-subspace-m5-f2?representation=markdown
- Record maturity: Prospecting problem with recorded work

## The problem

Determine the maximum dimension of an \(\mathbb F_2\)-linear subspace \(V\subseteq M_5(\mathbb F_2)\) such that every element of \(V\) is nilpotent and \(AB=BA\) for all \(A,B\in V\). Classify every maximum-dimensional such subspace up to conjugation by \(GL_5(\mathbb F_2)\).

### Definitions

- **Definition.** A matrix is nilpotent if some positive power of it is the zero matrix.
- **Definition.** Two subspaces V and W are conjugate when W=gVg^{-1} for some invertible matrix g.

### What counts as a solution

- Prove the maximum possible dimension and exhibit a subspace attaining it.
- Give one canonical representative of every GL5(F2)-orbit at the maximum dimension, with a complete orbit or stabilizer certificate.

## Status

**Resolution status:** Open.

**Packet-reported status.**

As of 2026-08-01, Britnell and Wildon classify commuting nilpotent similarity classes, while the maximum-dimensional linear-subspace orbit classification over F_2 remains unverified. The unresolved target is: Prove the maximum possible dimension and exhibit a subspace attaining it. Give one canonical representative of every GL5(F2)-orbit at the maximum dimension, with a complete orbit or stabilizer certificate. [1](#reference-1)

**Recorded scope.** Scope statement unavailable. Read the exact record before reusing this result.

[Read this exact record and its supporting evidence](/records/commuting-nilpotent-subspace-m5-f2-status-20260801)


*This research-packet status has not passed editorial review.*

## Research packet

### Evidence for the packet-reported status

**Claim 1 (Current checked status and unresolved remainder).** As of 2026-08-01, Britnell and Wildon classify commuting nilpotent similarity classes, while the maximum-dimensional linear-subspace orbit classification over F_2 remains unverified. The unresolved target is: Prove the maximum possible dimension and exhibit a subspace attaining it. Give one canonical representative of every GL5(F2)-orbit at the maximum dimension, with a complete orbit or stabilizer certificate.

A dated review on 2026-08-01 checked the structured sources below, exact-title and exact-parameter searches, nearby formulations, and the current TheoremDB corpus. Britnell and Wildon classify commuting nilpotent similarity classes, while the maximum-dimensional linear-subspace orbit classification over F_2 remains unverified.

A complete resolution must satisfy: Prove the maximum possible dimension and exhibit a subspace attaining it. Give one canonical representative of every GL5(F2)-orbit at the maximum dimension, with a complete orbit or stabilizer certificate.

### Background and intake notes

Maximizers have finite canonical descriptions. Rejected extensions, stabilizers, and common invariant flags can support later classifications over other fields and dimensions.

- Original intake status: UNKNOWN as of 2026-07-27. Dimension bounds for commuting and nilpotent matrix families are classical, but the exact GL5(F2)-orbit classification requested here was not located.
- A 2026-07-27 search checked Britnell and Wildon, arXiv:1001.0811, and exact searches for maximal commuting nilpotent subspaces of M_5(F_2). No fixed-field orbit list was located.
- Dimension five is the first odd size where the square-zero block construction has unequal block dimensions 2 and 3. The finite base field keeps orbit certification exact.
- Encode a subspace by a reduced row-echelon basis in the 25-dimensional matrix space. Check nilpotence on all 2^d elements and commutation on basis pairs, then quotient by GL5 conjugation.
- Trap: a basis of nilpotent matrices need not span only nilpotent matrices. Every linear combination must be checked or covered by a structural proof.

- Recorded example: Choose F_2^5=U direct-sum W with dim(U)=2 and dim(W)=3. The six-dimensional space of maps that kill U and send W into U has product zero, so all its elements are nilpotent and commute.

### Open directions

- **Route 1** (reported): Prove the maximum possible dimension and exhibit a subspace attaining it. Give one canonical representative of every GL5(F2)-orbit at the maximum dimension, with a complete orbit or stabilizer certificate. [1](#reference-1)

### Computational notes

- The block construction contains 2^6=64 matrices. Exact block multiplication verifies AB=0 for every two matrices in the space, giving the lower bound d>=6.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `commuting-nilpotent-subspace-m5-f2`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>John R. Britnell and Mark Wildon, “On types and classes of commuting matrices over finite fields,” Journal of the London Mathematical Society 83 (2011), 470–492; arXiv:1001.0811v1. Sections on commuting nilpotent similarity classes over finite fields. https://arxiv.org/abs/1001.0811
   - Also cited at Dataset references and 2026-08-01 exact-parameter status search.
   - Also cited at Editorial research route recorded 2026-08-01.
   - preprint; reference source; arXiv:1001.0811v1; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - Supplies the closest checked primary classification theory, while leaving the requested classification of commuting nilpotent linear subspaces of M_5(F_2) unverified.
   - Source named by the research packet.