Green functors and \(G\)-sets
Let \(G\) be a finite group and \(R\) a commutative ring. This book deals with Mackey and Green functors for \(G\) over \(R\), and with (left) modules over a Green functor \(A\). These are the objects of categories denoted by \({\mathcal M}ack(G)\), \({\mathcal G}reen(G)\) and \(A\)-\({\mathcal M}od\), respectively. Given \(A\), there is a useful category \({\mathcal C}_A\) whose objects are the finite \(G\)-sets and whose morphisms are given by \(\Hom_{{\mathcal C}_A}(X,Y)= A(Y\times X)\). An important property of \({\mathcal C}_A\) is the fact that the category \(A\)-\({\mathcal M}od\) is equivalent to the category \({\mathcal F}unct_R({\mathcal C}_A)\) of all representations of \({\mathcal C}_A\) (whose objects are the \(R\)-linear functors \({\mathcal C}_A\to R\)-\({\mathcal M}od\)). There is a third category which is equivalent to \(A\)-\({\mathcal M}od\) and \({\mathcal F}unct_R({\mathcal C}_A)\). This is the category \({\mathcal E}nd_{{\mathcal C}_A}(\Omega)\)-\({\mathcal M}od\) \((=A(\Omega^2)\)-\({\mathcal M}od)\) where \(\Omega\) is the finite \(G\)-set which is the disjoint union of all \(G\)-sets \(G/H\) where \(H\) ranges over all subgroups of \(G\). The \(R\)-algebra \(A(\Omega^2)\) can be described by generators and relations. It generalizes the Mackey algebra \(\mu_R(G)\) introduced by Thévenaz and Webb (where \(A\) is the Burnside ring Green functor \(b\)) and an algebra defined by Yoshida (where \(A\) is the fixed point functor \(FP_R\) and \(A\)-\({\mathcal M}od\) is the category of all cohomological Mackey functors). The category \(A\)-\({\mathcal M}od\) can also be investigated via functors \({\mathcal H}_A(-,-): A\)-\({\mathcal M}od\times A\)-\({\mathcal M}od\to {\mathcal M}ack(G)\) and \(-\otimes_A-:{\mathcal M}od\)-\(A\times A\)-\({\mathcal M}od\to {\mathcal M}ack(G)\) \((=b\)-\({\mathcal M}od)\). The functors \({\mathcal H}_A(-,-)\) and \(-\otimes_A-\) share the usual properties of hom and tensor functors. They can be used to develop a Morita theory for modules and bimodules over Green functors. In order to compare Mackey functors for different groups, one has induction, restriction, inflation and coinflation. More generally, for arbitrary finite groups \(G\) and \(H\), every \(G\)-\(H\)-biset \(U\) defines functors \(M\mapsto M\circ U\) from \({\mathcal M}ack(G)\) to \({\mathcal M}ack(H)\). The author constructs functors \({\mathcal L}_U,{\mathcal R}_U:{\mathcal M}ack(H)\to{\mathcal M}ack(G)\) which are left and right adjoint to \(M\mapsto M\circ U\). If \(A\) and \(B\) are Green functors for \(G\) and \(H\) then \(A\circ U\) and \({\mathcal L}_U(B)\) are Green functors for \(H\) and \(G\), respectively, and if \(N\) is a \(B\)-module then \({\mathcal L}_U(N)\) and \({\mathcal R}_U(N)\) are \({\mathcal L}_U(B)\)-modules. For an arbitrary Green functor \(A\), the simple \(A\)-modules can be parametrized by \(G\)-conjugacy classes of pairs \((H,V)\) where \(H\) is a subgroup of \(G\) and \(V\) is a simple \(\widehat A(H)\)-module; here \(\widehat A(H)\) is the skew group algebra \(\overline A(H)\otimes\overline N_G(H)\) where \(\overline N_G(H)= N_G(H)/H\) and \(\overline A(H)= A(H)/\sum_{K<H} t^H_K(A(K))\). This result implies Thévenaz's parametrization of simple Green functors and the Thévenaz-Webb parametrization of simple Mackey functors. The author also defines the center of a Green functor, in two different ways. Both can be used to obtain decompositions of Green functors.
scientific article; zbMATH DE number 1077996