Cell Modules for the Temperley-Lieb Algebra in Mixed Characteristic

Abstract: We study the representation theory of the Temperley-Lieb algebra $\text{TL}^k_n(\delta)$ in mixed characteristic, i.e. over an arbitrary field $k$ of characteristic $p$ and where $\delta$ satisfies some minimal polynomial $m_\delta$. In particular, we completely describe the submodule structure of cell modules for $\text{TL}_n$ and give their Alperin diagrams. The proof is entirely diagrammatic and does not appeal to the role of $\text{TL}_n$ as the endomorphism algebra of tensor powers of the fundamental representation of $\text{U}_q(\mathbf{sl}_2)$. We also investigate two-dimensional Jantzen-like filtrations of the cell modules related to the mixed characteristic.

 @misc{senecalmartinspencer2026cellmodules,
 title={Cell modules for the Temperley-Lieb algebra in mixed characteristic},
 author={Stuart Martin and Charles Senécal and Robert A. Spencer},
 year={2026},
 eprint={2601.17445},
 archivePrefix={arXiv},
 primaryClass={math.RT},
 url={https://arxiv.org/abs/2601.17445},
 }