The $6\times6$ equality case of matrix spaces with rank-two commutators

Abstract

Let $\mathcal V\subseteq M_6(\mathbb C)$ be a $17$-dimensional linear subspace such that $ \operatorname{rank}[S,T]\leq2 \quad(S,T\in\mathcal V). $ We prove that $\mathcal V$, or its transpose, is conjugate to the algebra $ \left\{ \begin{pmatrix} A&B&C\\ 0&λI_2&D\\ 0&0&λI_2 \end{pmatrix}: A,B,C,D\in M_2(\mathbb C),\ λ\in\mathbb C \right\}. $ Consequently, the corresponding closed algebraic locus in $\operatorname{Gr}(17,M_6(\mathbb C))$ is the disjoint union of two nonsingular irreducible components, each isomorphic to $\operatorname{Fl}(2,4;6)$. We also prove that the Zariski tangent space at $\mathcal A$ of the corresponding closed algebraic locus is equal to the tangent space to the conjugacy orbit of $\mathcal A$.

Disclosure

“RANK-TWO COMMUTATORS IN M6 (C) 13 Declaration of AI use This manuscript was developed with extensive use of OpenAI’s ChatGPT. Chat- GPT played a major role in the selection and refinement of the research problem, the iterative formulation of the main theorem, literature searches, exploration of proof strategies, development and assembly of the proof, verificatio”

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Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-22T18:21:57.156979Z

Structural counts