Cell modules for the Temperley-Lieb algebra in mixed characteristic

Abstract

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

31 pages, comments welcome