Modular Valenced Temperley-Lieb Algebras

TL;DR

This paper studies the representation theory of valenced Temperley-Lieb algebras across different characteristics, providing diagrammatic methods to determine their cellular structure, module dimensions, and indecomposables.

Contribution

It introduces a diagrammatic approach to analyze the cellular properties and module structures of valenced Temperley-Lieb algebras in various characteristics.

Findings

Determined cell indices and module dimensions for a wide class of algebras.

Developed a general framework for bases of cell modules.

Provided formulas for module dimensions and indecomposables.

Abstract

We investigate the representation theory of the valenced Temperley-Lieb algebras in mixed characteristic. These algebras, as described in characteristic zero by Flores and Peltola, arise naturally in statistical physics and conformal field theory and are a natural deformation of normal Temperley-Lieb algebras. In general characteristic, they encode the fusion rules for the category of $U_q(\mathfrak{sl}_2)$ tilting modules. We use the cellular properties of the Temperley-Lieb algebras to determine those of the valenced Temperley-Lieb algebras. Our approach is, at heart, entirely diagrammatic and we calculate cell indices, module dimensions and indecomposable modules for a wide class of valenced Temperley-Lieb algebras. We present a general framework for finding bases of cell modules and a formula for their dimensions.