Author: Peter Webb

Title:
Stratifications and Mackey Functors I: Functors for a Single Group

Address:
School of Mathematics,
University of Minnesota,
Minneapolis MN 55455,
USA

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 previously used 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.

Preprint: September 1999
Journal:  Proc. London Math. Soc. (3) 82 (2001), 299-336.