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PreprintCC BY 4.0arXiv

On a conjecture of King-Tollu-Toumazet for rectangular Kostka coefficients

  • Deepanshu Prasad
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Abstract

A conjecture of King, Tollu and Toumazet predicts the behavior of small Littlewood--Richardson coefficients under stretching: if cνλ,μ=3, then cMνMλ,Mμ equals 2M+1 or binomM+22 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 Kλ,a. We realize them as the weight multiplicities of the homogeneous coordinate ring of the Grassmannian 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 Gr(2,4), Gr(2,5) and 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 Gr(2,4), and binomM+22 for Gr(2,5) and Gr(3,5).

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Cite

BibTeXRIS
BibTeX
@misc{prasad2026conjecture,
  title = {{On a conjecture of King-Tollu-Toumazet for rectangular Kostka coefficients}},
  author = {Deepanshu Prasad},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21921},
  eprint = {2610.21921},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21921},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On a conjecture of King-Tollu-Toumazet for rectangular Kostka coefficients
AU  - Deepanshu Prasad
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21921
UR  - https://arxiv.org/abs/2610.21921
AB  - 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 binomM+22 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,a. We realize them as the weight multiplicities of the homogeneous coordinate ring of the Grassmannian 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 Gr(2,4), Gr(2,5) and 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 Gr(2,4), and binomM+22 for Gr(2,5) and Gr(3,5).
ER  -