Higher Jones Algebras and their simple Modules

TL;DR

This paper studies higher Jones algebras derived from tilting modules of reductive algebraic groups over fields of positive characteristic, determining their simple modules and dimensions, and applying these methods to various well-known algebra classes.

Contribution

It introduces a unified approach to determine simple modules and their dimensions for higher Jones algebras and related classes, including quantized analogues.

Findings

Simple modules and dimensions for higher Jones algebras are explicitly determined.

The approach applies to group algebras of symmetric groups, Brauer, Temperley--Lieb, Hecke, and BMW algebras.

Several detailed examples illustrate the method's effectiveness.

Abstract

Let $G$ be a connected reductive algebraic group over a field of positive characteristic $p$ and denote by $\mathcal T$ the category of tilting modules for $G$. The higher Jones algebras are the endomorphism algebras of objects in the fusion quotient category of $\mathcal T$. We determine the simple modules and their dimensions for these semisimple algebras as well as their quantized analogues. This provides a general approach for determining various classes of simple modules for many well-studied algebras such as group algebras for symmetric groups, Brauer algebras, Temperley--Lieb algebras, Hecke algebras and $BMW$-algebras. We treat each of these cases in some detail and give several examples.