{
  "abstract": "Let $0\\leq k < n$, and let $\\mathcal V\\subseteq M_n(\\mathbb C)$ be a complex linear subspace satisfying $\\operatorname{rank}[S,T]\\leq k$ for all $S,T\\in\\mathcal V$. Omladič, Radjavi, and Šivic proved the sharp bound $\\dim\\mathcal V\\leq nk+\\left\\lfloor (n-k)^2/4\\right\\rfloor+1$ 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 $n-k\\geq4$, 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 $2\\leq k\\leq n-2$, we determine all of its irreducible components. If $n-k\\geq4$, 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 $n-k\\geq4$, 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27928",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhi-Lin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fz84me8tr12p9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27928",
  "fulltext_url": null,
  "id": "fz84me8tr12p9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27928",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.27928",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fz84me8tr12p9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27928",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27928"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Equality cases for matrix spaces with bounded-rank commutators",
  "title_text": "Equality cases for matrix spaces with bounded-rank commutators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/fz84me8tr12p9",
  "venue": "arXiv",
  "versions": []
}