Equality cases for matrix spaces with bounded-rank commutators

Abstract

Let $0\le k<n$, and let $\mathcal{V}\subseteq M_n(\mathbb{C})$ be a complex linear subspace satisfying $\mathrm{rank}[S,T]\le k$ for all $S,T\in \mathcal{V}$. Omladič, Radjavi, and Šivic proved the sharp bound $\dim \mathcal{V}\le nk+\lfloor (n-k{)}^2/4\rfloor +1$ and conjectured a classification of the equality cases. We prove their conjecture. If equality holds, then, after a similarity and possibly transposition, $\mathcal{V}$ consists of all block upper-triangular matrices with arbitrary upper-left and upper-right blocks and with lower-right block in a maximal-dimensional commuting subspace of $M_{n-k}(\mathbb{C})$. For $n-k\ge 4$, these commuting subspaces are the classical equality cases in Schur's theorem; in dimensions $2$ and $3$, the additional equality cases also occur. At the equality dimension, the rank condition defines a projective algebraic subset of a Grassmannian. For $2\le k\le n-2$, we determine all of its irreducible components. If $n-k\ge 4$, there are exactly two components when $n-k$ is even and exactly four when $n-k$ is odd; if $n-k\in \{2,3\}$, there are exactly two. When $n-k\ge 4$, the Zariski tangent space at each such space equals the tangent space to its conjugacy orbit. When $n-k\in \{2,3\}$, the two components are obtained by varying the invariant $k$-dimensional subspace and the maximal-dimensional commuting subspace on the quotient, together with their transposes.

24 pages