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PreprintCC BY 4.0arXiv

A Note on Additive Diameter

  • Ernesto Ingrosso
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Abstract

We record two results on additive diameters of group representations. First we give a corrected version of Proposition 5.3 from the first arXiv version of arXiv:2504.07573. Let 0<ε<1/3, let n>9/ε2, and let U≤sln(ℂ) with dim U>ε n2. Then diamSLn(ℂ)+(sln(ℂ),U)≤ 32/ε+8. The proof uses an averaging argument for the action of the symmetric group on the off-diagonal matrix positions. This argument was subsequently developed further in arXiv:2609.03882. We also prove the right-hand inequality in Question 6.7 of arXiv:2504.07573. If G is a complex algebraic group, V is a finite-dimensional G-module, U≤ V, and g=Lie(G), then diamG+(V,U)≤diamg+(V,U). No irreducibility or connectedness assumption is needed.

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Cite

BibTeXRIS
BibTeX
@misc{ingrosso2026note,
  title = {{A Note on Additive Diameter}},
  author = {Ernesto Ingrosso},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06023},
  eprint = {2609.06023},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06023},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Note on Additive Diameter
AU  - Ernesto Ingrosso
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06023
UR  - https://arxiv.org/abs/2609.06023
AB  - We record two results on additive diameters of group representations. First we give a corrected version of Proposition 5.3 from the first arXiv version of arXiv:2504.07573. Let 0<ε<1/3, let n>9/ε^2, and let U≤sl_n(ℂ) with dim U>ε n^2. Then diam^SL_n(ℂ)_+(sl_n(ℂ),U)≤ 32/ε+8. The proof uses an averaging argument for the action of the symmetric group on the off-diagonal matrix positions. This argument was subsequently developed further in arXiv:2609.03882. We also prove the right-hand inequality in Question 6.7 of arXiv:2504.07573. If G is a complex algebraic group, V is a finite-dimensional G-module, U≤ V, and g=Lie(G), then diam^G_+(V,U)≤diam^g_+(V,U). No irreducibility or connectedness assumption is needed.
ER  -