First Author: Jacques Th\'evenaz
e-mail: Jacques.Thevenaz@ima.unil.ch
Address: Institut de Math\'ematiques,
Universit\'e de Lausanne,
CH--1015 Lausanne,
Switzerland
Second Author: Peter Webb
email: webb@math.umn.edu
Address:
School of Mathematics,
University of Minnesota,
Minneapolis MN 55455
Title:
The structure of Mackey functors
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.
Journal:
Trans. A.M.S. 347(1995), 1865-1961.