The slice Burnside ring and the section Burnside ring of a finite group

Abstract

This paper introduces two new Burnside rings for a finite group G, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of G-sets and of Galois morphisms of G-sets, respectively. The well-known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural Green biset functor structure. The functorial structure of unit groups of these rings is also discussed.

Simplicial Burnside ring

“…It shares almost all properties of the Burnside ring. In particular, as already shown (see [3] for a more complete description), the slice Burnside ring is a commutative ring, which is free of finite rank as a Z-module. The investigation of the slices, that is the pairs of groups (T , S) such that S is a subgroup of T , is a central subject in the study of the slice Burnside ring.…”

The ideals of the slice Burnside p-biset functor

“…It is an analogue of the classical Burnside ring constructed from the morphisms of G-sets instead the G-sets themselves, and it shares most of its properties. In particular, as already shown by Serge Bouc (see [3] for more complete description), the slice Burnside ring is a commutative ring, which is free of finite rank as a Z-module, and it becomes a split semisimple Q-algebra, after tensoring with Q. The correspondence which assigns to each finite group its slice Burnside ring has a natural biset functor structure, for which it becomes a Green biset functor.…”

“…We first recall the definition and basic properties of the slice Burnside ring introduced in [3], to which we refer the reader for all statements without proof. • The set of slices of G is denoted by Π(G).…”

“…Theorem 3.5. [ [3], Theorem 3.9] Let G and H be finite groups, and let U be a finite (H, G)-biset. Then the correspondence…”

On the T∘-slices of a finite group

“…In [3], the slice Burnside ring (G) is introduced: it is a commutative ring, which has a Z-basis T , S G indexed by the conjugacy classes of slices (T , S) of G. It is shown that Q ⊗ (G) is a split semisimple Q-algebra, whose primitive idempotents are also indexed by conjugacy classes of slices of G, and given by…”