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 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.

Standard stratifications of EI categories and Alperin's weight conjecture

“…In [29,Theorem 9.1] we gave a sufficient condition for the Ringel dual algebra to be standardly stratified, but unfortunately there was a significant typographical error in the statement of the theorem, and we take the opportunity to point this out. In that theorem the modules which are here called Δ λ were called Θ(λ) and the hypothesis Hom(Θ(λ), Θ(ρ)) = 0 unless λ > ρ was made.…”

“…It should have read 'Hom(Θ(λ), Θ(ρ)) = 0 unless λ ρ,' and the proof used this latter condition. We comment also that in [29,Theorem 9.10] the Θ(λ) were indexed by the opposite preorder to the one used here.…”

“…The Ringel dual of a standardly-stratified algebra was constructed in [2] (under the hypothesis that Λ is a poset) and it was also constructed independently in [29,Theorem 9.1] (without this hypothesis), where as part of a more general construction it was shown that the Ringel dual of a standardly-stratified algebra is again standardly-stratified. The approach used in [29] was to classify the indecomposable Ext-injective objects in F (Δ) as the 'Ext-injective hulls' T λ of the Δ λ , without using the property that they are precisely the indecomposable objects in F (Δ) ∩ F (∇).…”

“…The approach used in [29] was to classify the indecomposable Ext-injective objects in F (Δ) as the 'Ext-injective hulls' T λ of the Δ λ , without using the property that they are precisely the indecomposable objects in F (Δ) ∩ F (∇).…”

“…In [29] the Ringel dual algebra was introduced using a preordered set, and it was shown there that the dual algebra is again standardly-stratified, but it was not shown that a standardly-stratified algebra may be identified with its double dual, and neither was Dlab's theorem considered. In fact all of these results hold in the generality that the simple modules are parametrized by a preordered set provided we are careful to set things up in the right way.…”