Equality-case classification for rank-bounded commutator spaces

Classify all equality cases for the dimension bound on complex linear subspaces \(\mathcal V\subseteq M_n(\mathbb C)\) satisfying \(\operatorname{rank}[S,T]\leq k\) for all \(S,T\in\mathcal V\), where the bound is \(\dim\mathcal V\leq nk+\left\lfloor (n-k)^2/4\right\rfloor+1\), as asserted by the Omladič–Radjavi–Šivic conjecture; the paper establishes only the case \((n,k)=(6,2)\).

Background

For a complex linear subspace of $M_n(\mathbb C)$ whose pairwise commutators have rank at most $k$, Omladič, Radjavi, and Šivic proved the dimension bound $nk+\lfloor (n-k)^2/4\rfloor+1$. They also formulated a conjectural classification of the spaces attaining equality. Prior results covered the cases $k=1$, $k=n-1$, and arbitrary $k$ under the additional assumption that the subspace is an algebra. The present paper proves the conjectured classification for the previously specified case $(n,k)=(6,2)$, leaving the general equality-case classification as the explicitly identified conjectural problem.