A remark on Mackey-functors

Lindner, Harald

doi:10.1007/bf01245921

Cited by 70 publications

(45 citation statements)

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“…Thus we shall also use freely this second approach. For completeness we mention also a third version of the definition of a Mackey functor which is due to Lindner [15] (see also [25]) and which is probably the most elegant one. Finally a module-theoretic approach will be developed in the next section.…”

Section: X3 -^ X4mentioning

confidence: 99%

The structure of Mackey functors

Thévenaz¹,

Webb²

1995

Trans. Amer. Math. Soc.

153

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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.

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Section: X3 -^ X4mentioning

confidence: 99%

The structure of Mackey functors

Thévenaz¹,

Webb²

1995

Trans. Amer. Math. Soc.

153

View full text Add to dashboard Cite

show abstract

“…We present two definitions here, the first in terms of many axioms and the second in terms of bivariant functors on the category of finite G-sets. They may also be defined as functors on a specially-constructed category, an approach which is due to Lindner [40].…”

Section: The Definitions Of a Mackey Functormentioning

confidence: 99%

The Structure of Mackey Functors

Thévenaz

Webb²

1995

Transactions of the American Mathematical Society

166

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“…This latter condition is omitted in this paper. The use of M k in the definition of the Mackey functor was first done by Linder [25] under a more general categorical setting. (ii) Let X be a free G-space.…”

Section: Functorsmentioning

confidence: 99%

“…However, our namings of the maps in Definition 2.2 fit more naturally with the definition which requires contravariance. [14,10,25,22,24] is essentially an additive functor from the Mackey category M k to the category of k-modules, which transforms finite sum in M k to finite sum in the category of k-modules, with the assumption that G is finite. This latter condition is omitted in this paper.…”

Section: Functorsmentioning

confidence: 99%