Construction of functors between categories of \(G\)-sets
type: Item
instance of: scholarly article
title: Construction of functors between categories of \(G\)-sets
zbMATH Open document ID: 0858.19002
author: Serge Bouc
published in: Journal of Algebra
publication date: 1997-01-23T00:00:00Z
review text: The paper under review contains a number of very substantial new ideas. Let \(G\) and \(H\) be finite groups. A \(G\)-set-\(H\) is a finite set \(A\) on which \(G\) acts from the left, \(H\) from the right, and both actions commute. Finite \(G\)-sets-\(H\) form a category \(G\)-\({\mathcal S}et\)-\(H\) with respect to equivariant maps. For a third group \(K\) and an \(H\)-set-\(K\) \(B\), the author defines a \(G\)-set-\(K\) \(A {\circ_H} B\). In this way a functor \(A {\circ_H} - :H\)-\({\mathcal S}et\)-\(K \to G\)-\({\mathcal S}et\)-\(K\) is obtained. In the special case \(K = 1\), \(A {\circ_H} -\) can be viewed as a functor \(H\)-\({\mathcal S}et\to G\)-\({\mathcal S}et\). This functor is shown to preserve disjoint unions and pullbacks. One of the main results in the paper shows that this yields a bijection between isomorphism classes of finite \(G\)-sets-\(H\) and isomorphism classes of functors \(F:H\)-\({\mathcal S}et\to G\)-\({\mathcal S}et\) preserving disjoint unions and pullbacks. One can form the Grothendieck ring \(\Gamma(G)\) of the category \(G\)-\({\mathcal S}et\)-\(G\) with respect to disjoint unions and the product \(\circ_G\). Given a family \({\mathcal F}\) of subgroups of \(G\) closed under conjugation and normal products, the author constructs idempotents \(E^G_P\) in \(\Gamma(G)\), for (conjugacy classes of) subgroups \(P\) in \(\mathcal F\). These idempotents form a decomposition of \(1 \in \Gamma(G)\) into pairwise orthogonal idempotents \(E^G_P \in \Gamma(G)\). The Burnside ring \(b(G)\) becomes a module over \(\Gamma (G)\) via \(\circ_G\), and the idempotents \(E^G_P\) become the projectors on \(b(G)\) considered in an earlier paper by the author [J. Algebra 139, No. 2, 395-445 (1991; Zbl 0735.19001)]. There is also a homomorphism of rings \(b(G) \to \Gamma(G)\). If \(\mathcal K\) is a field of characteristic zero then the scalar extension \(b_{\mathcal K} (G)\) of \(b(G)\) is known to be semisimple, and there is a corresponding decomposit…
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 923704
zbMATH Keywords: category of finitely generated \(R\)-modules
zbMATH Keywords: finite groups
zbMATH Keywords: actions
zbMATH Keywords: equivariant maps
zbMATH Keywords: isomorphism class of finite \(G\)-sets-\(H\)
zbMATH Keywords: functors
zbMATH Keywords: Grothendieck ring
zbMATH Keywords: Burnside ring
zbMATH Keywords: orthogonal primitive idempotents
zbMATH Keywords: permutation module
zbMATH Keywords: semisimple \(\mathcal K\)-algebra
zbMATH Keywords: Grothendieck group
zbMATH Keywords: additive functors
zbMATH Keywords: abelian category
zbMATH Keywords: indecomposable object
zbMATH Keywords: Mackey functors
zbMATH Keywords: group cohomology
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: \(p\)-local subgroups
zbMATH Keywords: Alperin's weight conjecture
zbMATH Keywords: generalized Steinberg modules
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: jabr.1996.0239
OpenAlex ID: W2015615038
DOI: 10.1006/JABR.1996.0239
description: scientific article; zbMATH DE number 923704