Burnside rings

Abstract

This chapter provides an overview of the basic properties, structure, and invariants of Burnside rings. The Burnside ring is the natural framework to study the invariants attached to structured G-sets. 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. The 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 leads to the Mackeyfunctor or Greenfunctor 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. This leads to a decomposition of the category of Mackey functors for G as a direct sum of smaller Abelian categories.