The Structure of Mackey Functors

Trace maps for Mackey algebras

“…The proof can be found in Proposition 4.5.1 of [1]. Let us recall that an explicit isomorphism β can be defined on the generators of μ R (G) by [15].) There is an equivalence of categories…”

“…Proof. Only for this proof, we use Green's definition of Mackey functors since it is much more convenient for understanding the action of the induction and restriction maps (see Section 2 of [15]). If (RB(H)) H G is a compatible family of symmetric algebras, then by Theorem 3.12, the Mackey algebra is symmetric.…”

“…In this paper we use the point of view of Dress who defined the Mackey functors as particular bivariant functors and we use the Mackey algebra introduced by Thévenaz and Webb. In [15] they proved that a Mackey functor is nothing but a module over the so-called Mackey algebra. Numerous properties of this algebra are known: this algebra shares a lot of properties with the group algebra.…”

“…For example, the Mackey algebra is a free R-module, and its R-rank doesn't depend on the ring R. If we work with a p-modular system which is "large enough", there is a decomposition theory, in particular the Cartan matrix of this algebra is symmetric. However there are some differences, over a field of characteristic p > 0, where p | |G|, the determinant of the Cartan matrix is not a power of the prime number p in general, and as shown in [15] the Mackey algebra is seldom a self-injective algebra. One may wonder about a stronger property for the Mackey algebra: when is the Mackey algebra a symmetric algebra?…”

“…Over a field of characteristic p > 0 which is "large enough", where p | |G|, then Jacques Thévenaz and Peter Webb proved that the so-called p-local Mackey algebra (see Section 2 of [2]) is selfinjective if and only if the Sylow p-subgroups of G are of order p (Theorem 19.2 of [15]). However, in the same article, they proved that the p-local Mackey algebra is a product of matrix algebras and Brauer tree algebras (Theorem 20.2 of [15]). Since a Brauer tree algebra is derived equivalent to a symmetric Nakayama algebra, then by [13] or, for a more general result [18], all Brauer tree algebras are symmetric algebras.…”