The structure of Mackey functors

Abstract

Abstract.Mackey functors are a framework having the common properties of many natural constructions for finite groups, such as group cohomology, representation rings, the Burnside ring, the topological K-theory of classifying spaces, the algebraic K-theory of group rings, the Witt rings of Galois extensions, etc. In this work we first show that the Mackey functors for a group may be identified with the modules for a certain algebra, called the Mackey algebra. The study of Mackey functors is thus the same thing as the study of the representation theory of this algebra. We develop the properties of Mackey functors in the spirit of representation theory, and it emerges that there are great similarities with the representation theory of finite groups. In previous work we had classified the simple Mackey functors and demonstrated semisimplicity in characteristic zero. Here we consider the projective Mackey functors (in arbitrary characteristic), describing many of their features. We show, for example, that the Cartan matrix of the Mackey algebra may be computed from a decomposition matrix in the same way as for group representations. We determine the vertices, sources and Green correspondents of the projective and simple Mackey functors, as well as providing a way to compute the Ext groups for the simple Mackey functors. We parametrize the blocks of Mackey functors and determine the groups for which the Mackey algebra has finite representation type. It turns out that these Mackey algebras are direct sums of simple algebras and Brauer tree algebras. Throughout this theory there is a close connection between the properties of the Mackey functors, and the representations of the group on which they are defined, and of its subgroups. The relationships between these representations are exactly the information encoded by Mackey functors. This observation suggests the use of Mackey functors in a new way, as tools in group representation theory.

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

“…Since the multiplicity of a simple functor as a composition factor in another functor M is determined by the modules M (G) as G varies (using the method of [32] and [31]), we see that the composition factors of ∆ H,V * are the duals of the composition factors of ∆ H,V , and these are the same as the composition factors of ∇ H,V . In fact, we see that to guarantee the symmetry of Cartan invariants such as…”

“…The approach we take is exactly the same as in [31,Section 14] and the arguments presented there go through here also. Theorem 5.10.…”

On the Cartan matrix of Mackey algebras

“…It is an associative algebra with the property that the category μ R (G)-Mod of left μ R (G)-modules is equivalent to the category Mack R (G) of Mackey functors for G over R. 2.4. Thévenaz and Webb have also shown ( [13], Theorem 10.1) that there is an equivalence of abelian categories…”

“…It has been shown by J. Thévenaz and P. Webb ( [13]), among many fundamental other results, that the category of Mackey functors for G over R is equivalent to the category of modules over the Mackey algebra μ R (G). This algebra shares many properties with the group algebra RG: it is free as an R-module, and its R-rank does not depend on R; if K is a field of characteristic 0 or coprime to the order of G, the algebra μ K (G) is semisimple; when (K, O, k) is a p-modular system, there is a decomposition theory from Mackey functors for G over K to Mackey functors for G over k; the Cartan matrix of μ k (G) is symmetric and nonsingular.…”

“…It was shown by Thévenaz and Webb (see [13], Theorem 17.1 and its proof; see also [4])) that the blocks of the algebra μ k (G, 1) and the blocks of the algebra coμ k (G) are in one-to-one correspondence with the blocks of the group algebra kG. (1) The rank of the Cartan matrix of the algebra coμ k (b) is equal to …”