Equality cases for matrix spaces with bounded-rank commutators
Abstract
Let 0≤ k < n, and let 𝒱⊆ Mn(ℂ) be a complex linear subspace satisfying rank[S,T]≤ k for all S,T∈𝒱. Omladič, Radjavi, and Šivic proved the sharp bound dim𝒱≤ nk+⌊ (n−k)2/4⌋+1 and conjectured a classification of the equality cases. We prove their conjecture. If equality holds, then, after a similarity and possibly transposition, 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 Mn−k(ℂ). For n−k≥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≤ k≤ n−2, we determine all of its irreducible components. If n−k≥4, there are exactly two components when n−k is even and exactly four when n−k is odd; if n−k∈{2,3}, there are exactly two. When n−k≥4, the Zariski tangent space at each such space equals the tangent space to its conjugacy orbit. When n−k∈{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.
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BibTeX
@misc{zhang2026equality,
title = {{Equality cases for matrix spaces with bounded-rank commutators}},
author = {Zhi-Lin Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27928},
eprint = {2609.27928},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27928},
}
RIS
TY - UNPB
TI - Equality cases for matrix spaces with bounded-rank commutators
AU - Zhi-Lin Zhang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27928
UR - https://arxiv.org/abs/2609.27928
AB - Let 0≤ k < n, and let 𝒱⊆ M_n(ℂ) be a complex linear subspace satisfying rank[S,T]≤ k for all S,T∈𝒱. Omladič, Radjavi, and Šivic proved the sharp bound dim𝒱≤ nk+⌊ (n−k)^2/4⌋+1 and conjectured a classification of the equality cases. We prove their conjecture. If equality holds, then, after a similarity and possibly transposition, 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(ℂ). For n−k≥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≤ k≤ n−2, we determine all of its irreducible components. If n−k≥4, there are exactly two components when n−k is even and exactly four when n−k is odd; if n−k∈{2,3}, there are exactly two. When n−k≥4, the Zariski tangent space at each such space equals the tangent space to its conjugacy orbit. When n−k∈{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.
ER -