{
  "abstract": "Let a group $G$ act linearly on a finite-dimensional complex vector space $V$. The group-additive diameter of a subspace $U \\leq V$ is the least number of translates of $U$ whose sum is all of $V$. Counting dimensions, it is at least $\\dim V / \\dim U$. We show that when $G$ is compact and $V$ is irreducible, the diameter of every nonzero subspace is at most $\\lceil (\\dim V / \\dim U) \\ln \\dim V \\rceil$, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity $\\mathsf{C}(V)$, the largest ratio between the diameter of any subspace and its trivial lower bound, so that $1 \\leq \\mathsf{C}(V) \\leq 2 + \\ln \\dim V$, and we determine where in this range various representations lie. Every irreducible representation of $\\mathrm{SL}_2(\\mathbf{C})$ has $\\mathsf{C}(V) = 1$. The logarithm can be genuinely present along families of symmetric and exterior powers of $\\mathrm{SL}_n(\\mathbf{C})$ with $n$ varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as $\\mathrm{PSL}_2(\\mathbf{F}_p)$. The complexity is bounded above by a constant on the conjugation representations of $\\mathrm{SL}_n(\\mathbf{C})$ and on the representations $\\operatorname{Sym}^k \\mathbf{C}^3$ of $\\mathrm{SL}_3(\\mathbf{C})$. On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra $\\mathfrak{sl}_3(\\mathbf{C})$ acting on $\\operatorname{Sym}^k \\mathbf{C}^3$, the monomial diameter with respect to $\\operatorname{Sym}^k X$ for a plane $X \\leq \\mathbf{C}^3$ is optimal, while the corresponding $\\mathrm{SL}_3(\\mathbf{C})$ diameter is not.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03882",
  "author_total": 2,
  "authors": [
    {
      "name": "Urban Jezernik"
    },
    {
      "name": "Špela Špenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4axxgfqq6bawg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03882",
  "fulltext_url": null,
  "id": "4axxgfqq6bawg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03882",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7208748655",
  "pdf_url": "https://arxiv.org/pdf/2609.03882",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4axxgfqq6bawg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03882",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03882"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Additive diameters and covering complexity of irreducible representations",
  "title_text": "Additive diameters and covering complexity of irreducible representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/4axxgfqq6bawg",
  "venue": "arXiv",
  "versions": []
}