[#P2748] Largest commuting nilpotent subspaces of M5(F2)
Contents
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 \(GL_5(\mathbb F_2)\).
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Definitions and notation
1Definitions
Definition 1. A matrix is nilpotent if some positive power of it is the zero matrix.
Definition 2. Two subspaces V and W are conjugate when W=gVg^{-1} for some invertible matrix g.
2What 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.
1Status
Saved packet · August 1, 2026
Packet-reported status (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.[1]
This research-packet status has not passed editorial review.
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Notes and companion material
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 1. 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.
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.
How the 2 records connect
ProblemLargest commuting nilpotent subspaces of M5(F2)
All 1 recorded relations between these records and the problem
2See also
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“Largest commuting nilpotent subspaces of M5(F2).” TheoremDB. P2748. Problem statement; statement text SHA-256 02e81f6ab6271df15515a99766d42edc5dc14ec98fc316c36acde3fd7c01f71f. https://theoremdb.org/statement/?ref=P2748
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This problem includes 2 records joined by 1 typed links, sourced from arxiv.org[1], current as of August 1, 2026.
1References
- Packet source. 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. ↗preprint · reference source · arXiv:1001.0811v1 · checked 2026-08-01Source use: citation only.Source used to assess the problem's recorded status.Also cited at Dataset references and 2026-08-01 exact-parameter status search.Also cited at Editorial research route recorded 2026-08-01.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.
CC0 orbit-classification problem for pairwise commuting nilpotent matrix spaces over F2.
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