Primitive idempotents of the Grothendieck ring of Mackey functors
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.
scientific article; zbMATH DE number 1202949