Cell modules for the Temperley-Lieb algebra in mixed characteristic

TL;DR

This paper explores the representation theory of the Temperley-Lieb algebra over fields of mixed characteristic, providing a complete description of submodule structures and diagrammatic proofs without relying on quantum group representations.

Contribution

It offers a comprehensive diagrammatic analysis of cell modules for the Temperley-Lieb algebra in mixed characteristic, including submodule structures and Jantzen-like filtrations, independent of quantum group frameworks.

Findings

Complete submodule descriptions of cell modules

Diagrammatic proofs avoiding quantum group methods

Analysis of Jantzen-like filtrations in mixed characteristic

Abstract

We study the representation theory of the Temperley-Lieb algebra $\mathsf{TL}_n^k(\delta)$ 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 $\mathsf{TL}_n$ 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 $\textbf{U}_q(\mathfrak{sl}_2)$. We also investigate two-dimensional Jantzen-like filtrations of the cell modules related to the mixed characteristic.