Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces

Webb, Peter

doi:10.1016/0022-4049(93)90030-w

Cited by 57 publications

(62 citation statements)

References 9 publications

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“…As explained in [32], the full subcategory of the category of spectra whose objects are wedge sums of summands of the the spectra (BG + ) ∧ p , is equivalent to a full subcategory of Mack all,1 Zp whose objects are certain projective functors, and this is the connection between Corollary 10.2 and Theorem 7.2. We apply the theory in Section 11.4 to compute the Cartan matrices both of [ Prior to all this in our exposition, we introduce in Section 3 a theory of relative projectivity for globally defined Mackey functors.…”

Section: Introductionmentioning

confidence: 95%

“…It is useful to define globally defined Mackey functors by means of axioms, and such a set of such axioms was given in [33], extending axioms given for more restrictive kinds of functors in [32]. We will use these axioms in Section 5 to show that the functors ∆ H,V we construct are indeed globally defined Mackey functors.…”

Section: Definitions Of Globally Defined Mackey Functors and Preliminmentioning

confidence: 99%

“…In [32] they were used to give a method for computing group cohomology, and also a new proof of the theorem of Benson-Feshbach and Martino-Priddy on stable decomposition of classifying spaces of finite groups. In [12,9] they were used in a fundamental way in the determination of the Dade group of endopermutation modules.…”

Section: Introductionmentioning

confidence: 99%

“…On page 278 of [18] these functors are introduced in the special case Y = 1, where they are called 'global Mackey functors'. These same functors with the condition Y = 1 appear in [28] where they are called 'functors with Mackey structure' and also in [32] where they are called 'inflation functors'. In the last reference the functors which arise when X = Y = 1 are called 'global Mackey functors'.…”

Section: Introductionmentioning

confidence: 99%

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Stratifications and Mackey Functors II: Globally Defined Mackey Functors

Webb¹

2010

J K-Theor

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Abstract. We describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.

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Section: Introductionmentioning

confidence: 95%

Section: Definitions Of Globally Defined Mackey Functors and Preliminmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

Webb¹

2010

J K-Theor

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“…Indeed, Webb proved an explicit formula giving the evaluation of simple functors (see Theorem 2.6 in [We1]). …”

Section: The Bifree Double Burnside Ringmentioning

confidence: 99%

Simple biset functors and double Burnside ring

Bouc

Stancu

Thévenaz

2013

Journal of Pure and Applied Algebra

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: Let G be a finite group and let k be a field. Our purpose is to investigate the simple modules for the double Burnside ring kB(G, G). It turns out that they are evaluations at G of simple biset functors. For a fixed finite group H, we introduce a suitable bilinear form on kB(G, H) and we prove that the quotient of kB(−, H) by the radical of the bilinear form is a semi-simple functor. This allows for a description of the evaluation of simple functors, hence of simple modules for the double Burnside ring.

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A General Theory of Canonical Induction Formulae

Boltje

1998

Journal of Algebra

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So far there exist three independent constructions of two different canonical versions of Brauer's induction theorem for complex characters due to V. Snaith, P. Symonds, and the author. ''Canonical'' in this context means functorial with respect to restrictions along group homomorphisms. In this article we axiomatize the situation in which the above canonical induction formulae are constructed. Mackey functors and related structures arise in this way naturally as a convenient language. This approach allows us to construct canonical induction formulae for arbitrary Mackey functors. In particular we obtain canonical induction formulae for the Brauer character ring, the group of projective characters, the ring of trivial source modules, and the ring of linear source modules. In most cases, it is not difficult to construct such formulae over the rational numbers. A much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers. We give a sufficient condition in the general framework of Mackey functors for a canonical induction formula to be integral. As an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings, as, for example, the span of linear characters in the case of the canonical Brauer induction formula. This will be used in a subsequent article in the case of Adams operations and Chern classes.

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Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces

Cited by 57 publications

References 9 publications

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

Simple biset functors and double Burnside ring

A General Theory of Canonical Induction Formulae

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