Primitive idempotents of the Grothendieck ring of Mackey functors
type: Item
instance of: scholarly article
title: Primitive idempotents of the Grothendieck ring of Mackey functors
zbMATH Open document ID: 0914.20009
author: Fumihito Oda
published in: Hokkaido Mathematical Journal
publication date: 1999-06-17T00:00:00Z
review text: Let \(G\) be a finite group and \(R\) a commutative ring. A Mackey functor \(M\) for \(G\) over \(R\) consists of a family of \(R\)-modules \(M(H)\), one for each subgroup \(H\) of \(G\), together with certain induction, restriction and conjugation maps. These maps satisfy a number of axioms including the Mackey formula. With morphisms of Mackey functors properly defined, the Mackey functors for \(G\) over \(R\) form an abelian category \(\text{Mack}_R(G)\). One can also define a tensor product of Mackey functors in such a way that there is a corresponding Grothendieck ring \(G_0(\text{Mack}_R(G))\). The author constructs, in the case where \(R\) is a splitting field of characteristic 0 for \(N_G(H)/H\), for every subgroup \(H\) of \(G\), an isomorphism between \(G_0(\text{Mack}_R(G))\) and \(\bigoplus_HG_0(R[N_G(H)/H])\) where \(H\) runs through a transversal for the conjugacy classes of subgroups of \(G\). From this the primitive idempotents of \(G_0(\text{Mack}_R(G))\) can be obtained.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 16E20
zbMATH DE Number: 1202949
zbMATH Keywords: finite groups
zbMATH Keywords: induction
zbMATH Keywords: restriction
zbMATH Keywords: Mackey functors
zbMATH Keywords: Abelian categories
zbMATH Keywords: tensor products
zbMATH Keywords: Grothendieck rings
zbMATH Keywords: conjugacy classes of subgroups
zbMATH Keywords: primitive idempotents
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1202949