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 T the category of tilting modules for G. The higher Jones algebras are the endomorphism algebras of objects in the fusion quotient category of 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.