STRATIFICATIONS AND MACKEY FUNCTORS I: FUNCTORS FOR A SINGLE GROUP

Abstract

In the context of Mackey functors we introduce a category which is analogous to the category of modules for a quasi-hereditary algebra which have a filtration by standard objects. Many of the constructions which work for quasi-hereditary algebras can be done in this new context. In particular, we construct an analogue of the `Ringel dual', which turns out here to be a standardly stratified algebra. The Mackey functors which play the role of the standard objects are constructed in the same way as functors which have been used previously in parametrizing the simple Mackey functors, but instead of using simple modules in their construction (as was done before) we use p-permutation modules. These Mackey functors are obtained as adjoints of the operations of forming the Brauer quotient and its dual. The filtrations which have these Mackey functors as their factors are closely related to the filtrations whose terms are the sum of induction maps from specified subgroups, or are the common kernel of restriction maps to these subgroups. These latter filtrations appear in Conlon's decomposition theorems for the Green ring, as well as in other places, where they arise quite naturally.

2000 Mathematics Subject Classification: primary 20C20; secondary 20J05, 19A22, 16G70, 16E60.