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Cellular Algebra of A2 webs

  • Azat M. Gainutdinov
  • Jesper Lykke Jacobsen
  • Yueheng Li
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Abstract

We introduce a diagram algebra 𝒦L based on Kuperberg's A2-webs and prove a Schur--Weyl duality 𝒦L β‰… EndUq(sl3)((V+ βŠ• V-)βŠ— L), where VΒ± are the fundamental representations of Uq(sl3). Thus, 𝒦L is a natural A2 analogue of the Temperley--Lieb algebra. We construct an explicit cellular structure for 𝒦L, with the web basis as its cellular basis and cell modules indexed by dominant Uq(sl3)-weights. The bases of these modules are parametrized by dominant lattice paths via a recursive growth algorithm. Using this structure, we determine local generators of 𝒦L and establish web formulations of the first and second fundamental theorems of invariant theory. In particular, we show that the braid-group image, augmented by local idempotents, does not generate 𝒦L, and identify an additional local generator. Finally, we derive a bimodule decomposition of (V+ βŠ• V-)βŠ— L, yielding a complete set of irreducible 𝒦L-modules for formal or generic q.

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Cite

BibTeXRIS
BibTeX
@misc{gainutdinov2026cellular,
  title = {{Cellular Algebra of $A_2$ webs}},
  author = {Azat M. Gainutdinov and Jesper Lykke Jacobsen and Yueheng Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18801},
  eprint = {2610.18801},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18801},
}
RIS
TY  - UNPB
TI  - Cellular Algebra of A_2 webs
AU  - Azat M. Gainutdinov
AU  - Jesper Lykke Jacobsen
AU  - Yueheng Li
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18801
UR  - https://arxiv.org/abs/2610.18801
AB  - We introduce a diagram algebra 𝒦_L based on Kuperberg's A_2-webs and prove a Schur--Weyl duality 𝒦_L β‰… End_U_q(sl_3)((V^+ βŠ• V^-)^βŠ— L), where V^Β± are the fundamental representations of U_q(sl_3). Thus, 𝒦_L is a natural A_2 analogue of the Temperley--Lieb algebra. We construct an explicit cellular structure for 𝒦_L, with the web basis as its cellular basis and cell modules indexed by dominant U_q(sl_3)-weights. The bases of these modules are parametrized by dominant lattice paths via a recursive growth algorithm. Using this structure, we determine local generators of 𝒦_L and establish web formulations of the first and second fundamental theorems of invariant theory. In particular, we show that the braid-group image, augmented by local idempotents, does not generate 𝒦_L, and identify an additional local generator. Finally, we derive a bimodule decomposition of (V^+ βŠ• V^-)^βŠ— L, yielding a complete set of irreducible 𝒦_L-modules for formal or generic q.
ER  -