Cell modules for the Temperley-Lieb algebra in mixed characteristic

TL;DR

This work develops a completely diagrammatic, mixed-characteristic study of Temperley–Lieb algebras by describing the submodule structure of cell modules through Alperin diagrams and wiring these results to $(\ell,p)$-Jones–Wenzl theory. It introduces the $(\ell,p)$-digits framework, reconstructs the cellular bases via light ladders in the TL category, and proves that the submodule lattice of $\mathcal{S}(n,m)$ is governed by up-admissible sets, with truncation functors preserving these filtrations. The paper further develops a two-dimensional Jantzen-like filtration on cell modules, computed from Gram matrices and the mixed-characteristic data, revealing a rich, fractal-like modular structure in the composition factors and their placements. These diagrammatic methods illuminate the interaction between quantum numbers, mixed characteristic phenomena, and Kazhdan–Lusztig-type filtrations, offering tools to analyze forms, filtrations, and decomposition in TL algebras beyond characteristic zero.

Abstract

We study the representation theory of the Temperley-Lieb algebra $\mathsf{TL}_n^k(δ)$ in mixed characteristic, i.e. over an arbitrary field $k$ of characteristic $p$ and where $δ$ satisfies some minimal polynomial $m_δ$. 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.

Figures (3)

• Figure 1: Values of $\text{supp}(n)$ for $\ell=5$ and $p=3$. The $y$-axis is $n$ and a square at $(x,y)$ is coloured iff $x \in \text{supp}(y)$.
• Figure 2: The down-admissible sets for $685=[1,2,0,0,2,1]_{5,3}-1$ (right) and some of the up-admissible sets for $123 = [2,2,0,4]_{5,3}-1$ (left). The bars are aligned at the set ${0,1,2,3,4}$ which is common to both.
• Figure 3: The $42$ composition factors of $\mathcal{S}(10 000,18)$.

Theorems & Definitions (93)

• Definition 1.1
• Lemma 1.2
• Lemma 1.3
• proof
• Lemma 1.4
• Lemma 1.5
• proof
• Remark 1.6
• Lemma 1.7
• proof