{
  "abstract": "A conjecture of King, Tollu and Toumazet predicts the behavior of small Littlewood--Richardson coefficients under stretching: if $c^ν_{λ,μ}=3$, then $c^{Mν}_{Mλ,Mμ}$ equals $2M+1$ or $\\binom{M+2}{2}$ for every positive integers $M$. This corresponds to the Hilbert functions of a conic, or of the projective plane. We prove their conjecture for all rectangular Kostka coefficients $K_{λ,\\mathbf{a}}$. We realize them as the weight multiplicities of the homogeneous coordinate ring of the Grassmannian $\\mathrm{Gr}(k,n)$, or equivalently the weight spaces of semi-invariants of the $n$-subspace quiver with dimension vector $β=(k,1,1,\\dotsc,1)$. For a weight $σ$, we show that if $β$ is $σ$-stable, then the dimension of the corresponding weight space is at least $n-2$ when $k=2$ and at least $4$ for every Grassmannian other than $\\mathrm{Gr}(2,4)$, $\\mathrm{Gr}(2,5)$ and $\\mathrm{Gr}(3,5)$. These three cases are precisely when the associated moduli space of $σ$-stable representations has dimension at most two. Via $σ$-stable decomposition, the two alternatives are determined as follows: $2M+1$ for $\\mathrm{Gr}(2,4)$, and $\\binom{M+2}{2}$ for $\\mathrm{Gr}(2,5)$ and $\\mathrm{Gr}(3,5)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21921",
  "author_total": 1,
  "authors": [
    {
      "name": "Deepanshu Prasad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fqghk50py2d4r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21921",
  "fulltext_url": null,
  "id": "fqghk50py2d4r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21921",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222500545",
  "pdf_url": "https://arxiv.org/pdf/2610.21921",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fqghk50py2d4r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21921",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21921"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On a conjecture of King-Tollu-Toumazet for rectangular Kostka coefficients",
  "title_text": "On a conjecture of King-Tollu-Toumazet for rectangular Kostka coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/fqghk50py2d4r",
  "venue": "arXiv",
  "versions": []
}