Stratifications and Mackey functors. I: Functors for a single group (Q2766393)

In this interesting paper, the author shows that ideas developed for the module category of a quasi-hereditary algebra can be carried over to the category of Mackey functors for a finite group \(G\). Given a Mackey functor \(M\) for \(G\) over a field \(R\) of characteristic \(p\), the author constructs two filtrations on \(M\); these filtrations depend upon a total ordering of the set of conjugacy classes of subgroups of \(G\) refining the partial order \(\leq_G\) where \(A\leq_GB\) if and only if \(A\) is conjugate in \(G\) to a subgroup of \(B\). Successive quotients in these filtrations are closely related to special Mackey functors \(\Delta_{H,U}\) and \(\nabla_{H,U}\) constructed from \(R[N_G(H)/H]\)-modules \(U\) for subgroups \(H\) of \(G\).NEWLINENEWLINENEWLINEThe author considers the category \(\mathcal D\) of Mackey functors having a filtration with factors isomorphic to Mackey functors \(\Delta_{H,U}\) where \(U\) is a \(p\)-permutation module. The category \(\mathcal D\) behaves well with respect to restriction, induction and direct summands. The author shows that \(\mathcal D\) has relative Auslander-Reiten sequences. The projective Mackey functors for \(G\) are contained in \(\mathcal D\). The author investigates the Ext-projective and the Ext-injective Mackey functors in \(\mathcal D\).NEWLINENEWLINENEWLINEThe endomorphism ring \(E\) of the direct sum of a complete set of representatives for the isomorphism classes of indecomposable Ext-injective objects in \(\mathcal D\) is ``standardly stratified'' in the sense of Cline-Parshall-Scott. It is similar to the Ringel dual of a quasi-hereditary algebra. Moreover, there is an idempotent \(e\) in \(E\) such that \(eEe\) is Morita equivalent to the group algebra \(RG\). The author expresses the hope that some of the ideas in this paper could be relevant in connection with Alperin's weight conjecture.