Stratifications and Mackey functors I: Functors for a single group

In the context of Mackey functors we introduce a category which is analogou s to the category of modules for a quasi-hereditary algebra which have a fi ltration by standard objects. Many of the constructions which work for quas i-hereditary algebras can be done in this new context. In particular, we co nstruct an analogue of the 'Ringel dual', which turns out here to be a stan dardly stratified algebra. The Mackey functors which play the role of the s tandard objects are constructed in the same way as functors which have been previously used in parametrizing the simple Mackey functors, but instead o f using simple modules in their construction las 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 whi ch have these Mackey functors as their factors are closely related to the f iltrations whose terms are the sum of induction maps from specified subgrou ps, 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.