Burnside rings

Let B be the Burnside ring considered as a globally-defined Mackey functor, and k be a field of positive characteristic q. We prove that, when k ⊗ℤ B is restricted to nilpotent groups of order prime to q, the only simple Mackey functors that can appear as subquotients of it are of the form SE,k, where E is the direct product of elementary abelian p-groups. In the special case k = 𝔽2, the simple Mackey functor SE,𝔽2 must appear as a filtration factor in 𝔽2 ⊗ℤ B for every direct product of elementary abelian p-groups E of odd order.

In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gr\"othendieck group.

In this paper we introduce the notion of a categorical Mackey functor. This categorical notion allows us to obtain new Mackey functors by passing to Quillen’s K-theory of the corresponding abelian categories. In the case of an action by monoidal autoequivalences on a monoidal category the Mackey functor obtained at the level of Grothendieck rings has in fact a Green functor structure.

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.

Control of fusion and cohomology of trivial source modules

Bredon homology and ramified covering G-maps

Classical algebraic K-theory: The functors K0, K1, K2

Composition Factors from the Table of Marks

Normalizing isomorphisms between Burnside rings

The Burnside ring \(\mathcal {B}(G)\) of a finite group G, a classical tool in group theory and representation theory, is studied from the point of view of computational commutative algebra. Starting from a table of marks, we describe efficient algorithms for computing a presentation, the image of the mark homomorphism, the prime ideals and the prime ideal graph, the singular locus, the conductor in its integral closure, the connected components of its spectrum, and its idempotents. On the way, we provide methods for identifying p-residual subgroups, direct products of subgroups of coprime order, commutator subgroups, and perfect subgroups.

We define Mackey functors over posets mimicking the classical notion and introduce a weak version of them. Then we show that they are acyclic by analyzing cofibrant and pseudo-projective objects in the category of functors indexed in a filtered poset. As application, we study homotopy colimits over posets and we give a homology decomposition for the classifying space of the Bianchi group Γ1.

Projective G-sets

Analogous results were proved later for KO, in the generality of compact Lie groups, by Atiyah and Segal [8], and for KFq, the algebraic K-theory spectrum associated to the finite field Fq, by Rector [29] using Quillen's [26] computation of iT*(KFq). In each case, the answer involves an appropriate representation of G, and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field (see [31]). If one considers cohomology theories constructed from other permutative categories, one expects to find analogous computations in terms of a completed representation ring of G in the given category, appropriately defined. In particular, stable cohomotopy, Ts*, is constructed from the category of finite sets [31], and for this category the analogue of the representation is a well-known object, the Burnside A(G) [13]. A(G) is a commutative with augmentation, so one may speak of A(G), the Burnside ring. Moreover, there is

Orthogonal units of the bifree double Burnside ring

The Burnside ring of fusion systems