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PreprintCC BY 4.0arXiv

Thickening realization and positivity properties of canonical bases

  • Jiepeng Fang
  • Xuhua He
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Abstract

Let U be a quantum group associated with a symmetric Cartan datum, let U̇ be its modified form, and let Ḃ be the canonical basis of U̇. Lusztig conjectured that the structure constants of the multiplication, comultiplication, and bilinear form in U̇ with respect to Ḃ belong to ℕ[v,v−1]. We introduce the \emph{thickening realization}, which relates Ḃ to the canonical basis of the negative part of a larger quantum group Ũ-. More precisely, it identifies the relevant structure constants in U̇ with the structure constants in Ũ-. As consequences, we prove, for arbitrary symmetric Cartan datum, Ḃ has the positivity properties for the comultiplication and bilinear form, as well as the positivity for the multiplication whenever one factor is spherical parabolic. In particular, Lusztig's conjecture holds for simply-laced finite type. We also prove the canonical bases of a broad class of tensor products of integrable modules have the positivity properties for the transition matrices and actions by Ḃ.

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Cite

BibTeXRIS
BibTeX
@misc{fang2026thickening,
  title = {{Thickening realization and positivity properties of canonical bases}},
  author = {Jiepeng Fang and Xuhua He},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06676},
  eprint = {2609.06676},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06676},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Thickening realization and positivity properties of canonical bases
AU  - Jiepeng Fang
AU  - Xuhua He
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06676
UR  - https://arxiv.org/abs/2609.06676
AB  - Let U be a quantum group associated with a symmetric Cartan datum, let U̇ be its modified form, and let Ḃ be the canonical basis of U̇. Lusztig conjectured that the structure constants of the multiplication, comultiplication, and bilinear form in U̇ with respect to Ḃ belong to ℕ[v,v^−1]. We introduce the \emph{thickening realization}, which relates Ḃ to the canonical basis of the negative part of a larger quantum group Ũ^-. More precisely, it identifies the relevant structure constants in U̇ with the structure constants in Ũ^-. As consequences, we prove, for arbitrary symmetric Cartan datum, Ḃ has the positivity properties for the comultiplication and bilinear form, as well as the positivity for the multiplication whenever one factor is spherical parabolic. In particular, Lusztig's conjecture holds for simply-laced finite type. We also prove the canonical bases of a broad class of tensor products of integrable modules have the positivity properties for the transition matrices and actions by Ḃ.
ER  -