Higher Jones Algebras and their simple Modules

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.