Burnside rings

1 Let G be a finite group. The Burnside ring B(G) of the group G is one of the fundamental representation rings of G, namely the ring of permutation representations. It is in many ways the universal object to consider when looking at the category of G-sets. It can be viewed as an analogue of the ring Z of integers for this category. It can be studied from different points of view. First B(G) is a commutative ring, and one can look at is prime spectrum and primitive idempotents. This leads to various induction theorems (Artin, Conlon, Dress): the typical statement here is that any (virtual) RG-module is a linear combination with suitable coefficients of modules induced from certain subgroups of G (cyclic, hypoelementary, or Dress subgroups). The Burnside ring is the natural framework to study the invariants attached to structured G-sets (such as G-posets, or more generally simplicial G-sets). Those invariants are generalizations for the category of G-sets of classical notions, such as the Möbius function of a poset, or the Steinberg module of a Chevalley group. They have properties of projectivity, which lead to congruences on the values of Euler-Poincaré characteristic of some sets of subgroups of G. The ring B(G) is also functorial with respect to G and subgroups of G, and this leads to the Mackey functor or Green functor point of view. There are close connections between the Burnside ring and the Mackey algebra. The Burnside Mackey functor is a typical example of projective Mackey functor. It is also a universal object in the category of Green functors. This leads to decomposition of the category of Mackey functors for G as a directs sum of smaller abelian categories. Finally B(G) is also functorial with respect to bisets, and this is leads to the definition of double Burnside rings. Those rings are connected to stable homotopy