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Atkinson-Lloyd conjecture for primitively intransitive spaces of linear operators

Let UU and VV be finite-dimensional vector spaces over a field F\mathbb{F}. Let S\mathcal{S} be a linear subspace of Hom(U,V)\mathrm{Hom}(U,V). We say that it is intransitive when Sx≠V\mathcal{S} x \neq V for all x∈Ux \in U, and we say that it is primitively intransitive when, in addition, there is no proper linear subspace WW of VV such that πV/WS\pi^{V/W} \mathcal{S} is an intransitive subspace of Hom(U,V/W)\mathrm{Hom}(U,V/W), where πV/W:V↠V/W\pi^{V/W} : V \twoheadrightarrow V/W stands for the canonical projection.

Atkinson and Lloyd have proved that if ∣F∣≥n|\mathbb{F}| \geq n and S\mathcal{S} is primitively intransitive then dim⁡S≤n(n−1)2\dim \mathcal{S} \leq \frac{n(n-1)}{2}. The conjecture states that the result holds without the cardinality assumption on F\mathbb{F}.

Progress summary

Open

No public proof or counterexample has been found, so the conjecture remains open when the underlying field is small.

Atkinson and Lloyd proved the dimension bound for primitively intransitive operator spaces when the field has at least nn elements; the conjecture asks whether that restriction can be removed. No posing date or resolution is recorded in the retrieved sources.

Known results

  • Atkinson and Lloyd: if ∣F∣≥n|\mathbb{F}|\geq n, then dim⁡S≤n(n−1)/2\dim\mathcal{S}\leq n(n-1)/2 for primitively intransitive spaces.

2025 related generalization

A 2025 preprint gives a broader Atkinson-type bound under a corresponding field-cardinality hypothesis; for d=1d=1 it recovers dim⁡S≤n(n−1)/2\dim\mathcal{S}\leq n(n-1)/2 when ∣F∣≥n|\mathbb{F}|\geq n, but it neither proves nor refutes the unrestricted conjecture.

Current status (as of August 2026): The cardinality-dependent bound is known, while the conjecture without the assumption ∣F∣≥n|\mathbb{F}|\geq n remains open.

Sources
Sources & referencesView supporting material

References

https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/primitive-spaces-of-matrices-of-bounded-rank/A2F2D9825F53DD6601C3A15DEFED391E

https://arxiv.org/abs/2508.06934

linear-algebra
Posted by Clément de Seguins Pazzis

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