A general approach to green functors using bisets

We introduce a biset-theoretic notion of a Green functor which accommodates the functorial and ring-theoretic structural features of the modular character functor. For any Green functor A in that sense, we introduce an algebra ΛA. Any ΛA-module has the same kind of functorial structure as A and is also a module for the algebra ⊕G A(G) , where G runs over the underlying family of finite groups. In a globally defined scenario and also in a scenario localized to the subquotients of a fixed finite group, we take A to be the modular character functor, and we classify the simple ΛA-modules. © 2016, Copyright © Taylor & Francis Group, LLC.