Construction of functors between categories of \(G\)-sets
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 decomposition of \(1 \in b_{\mathcal K} (G)\) into pairwise orthogonal primitive idempotents \(e^G_H \in b_{\mathcal K} (G)\), indexed by (conjugacy classes of) subgroups \(H\) of \(G\). This yields another decomposition of \(1 \in \Gamma_{\mathcal K} (G)\) into pairwise orthogonal idempotents \(\widetilde{e}^G_H \in \Gamma_{\mathcal K} (G)\). These two decompositions can be combined to get a third decomposition of \(1\in\Gamma_{\mathcal K}(G)\) into pairwise orthogonal idempotents \(F^G_{H,K} \in \Gamma_{\mathcal K} (G)\), indexed by (conjugacy classes of) pairs \((H,K)\) of subgroups of \(G\) such that \(K\) is normal in \(H\). For an arbitrary finite group \(S\), the sum of all idempotents \(F^G_{H,K}\) such that \(H/K\) is isomorphic to \(S\) turns out to be a central idempotent \(B_S\) in \(\Gamma_{\mathcal K} (G)\). This yields a decomposition of \(\Gamma_{\mathcal K} (G)\) into subrings \(B_S \Gamma_{\mathcal K} (G) B_S\), indexed by (isomorphism classes of) sections \(S\) of \(G\). The author shows that each subring \(B_S \Gamma_{\mathcal K} (G) B_S\) is isomorphic to \(\text{End}_{{\mathcal K} \text{Out}(S)} (V_S)\) where \(V_S\) is an explicitly known permutation module; in particular, \(\Gamma_{\mathcal K} (G)\) is a semisimple \(\mathcal K\)-algebra. Let \(\mathcal P\) and \(\mathcal Q\) be classes of finite groups closed under subgroups, factor groups and extensions. A finite \(G\)-set-\(H\) \(A\) is said to be \(\mathcal P\)-free-\(\mathcal Q\) if all left stabilizers \(G_a\) are in \(\mathcal P\) and all right stabilizers \(_a H\) are in \(\mathcal Q\). The author constructs the following additive category \({\mathcal C}({\mathcal P}, {\mathcal Q})\): The objects in \({\mathcal C} ({\mathcal P}, {\mathcal Q})\) are all finite groups, the set of morphisms \(G \to H\) is the Grothendieck group of finite \(\mathcal P\)-free-\(\mathcal Q\) \(G\)-sets-\(H\), and the composition of morphisms is given by \(\circ_H\). For an arbitrary ring \(R\), the additive functors from \({\mathcal C} ({\mathcal P}, {\mathcal Q})\) to the category \(R\)-\({\mathcal M}od\) of finitely generated \(R\)-modules form an abelian category \({\mathcal F}_R ({\mathcal P}, {\mathcal Q})\). By another paper of the author [J. Algebra 183, No. 3, 664-736 (1996; see the preceding review)] the simple objects in \({\mathcal F}_R ({\mathcal P}, {\mathcal Q})\) are parametrized, up to isomorphism, by pairs \((H,V)\) where \(H\) is a finite group and \(V\) is a simple \(\text{Out}(H)\)-module. The author shows that, when \(R\) is a field of characteristic zero and \({\mathcal P} = {\mathcal Q}\), the category \({\mathcal F}_R({\mathcal P}, {\mathcal Q})\) is semisimple in the sense that every indecomposable object is simple. One of the motivations to study the product \(\circ_G\) comes from the theory of Mackey functors. A Mackey functor for \(G\) can be defined as a certain bifunctor \(M : G\)-\({\mathcal S}et\to R\)-\({\mathcal M}od\). For a finite \(G\)-set-\(H\) \(A\), composition with the functor \(A {\circ_H}-:H\)-\({\mathcal S}et\to G\)-\({\mathcal S}et\) yields a Mackey functor \(H\)-\({\mathcal S}et\to R\)-\({\mathcal M}od\) which the author denotes by \(M \circ A\). An interesting example of this construction is obtained in the following way: Let \(M = H^i\) be the Mackey functor of group cohomology. Then, for a suitably chosen \(G\)-set-\(G\) \(c_G\), the Mackey functor \(M \circ c_G\) turns out to be the Mackey functor \(HH^i\) of Hochschild cohomology. The definition of \(M \circ A\) can be generalized to ``virtual'' \(G\)-sets-\(H\) \(A\). The result is a ``virtual'' Mackey functor \(M \circ A\). An example of this construction is the virtual Mackey functor \(N = FP_R \circ (c_G' - \text{Alp}_G)\). The restriction of this Mackey functor to an arbitrary subgroup \(H\) of \(G\) is relatively projective with respect to the family of all \(p\)-local subgroups of \(H\). Moreover, Alperin's weight conjecture turns out to be equivalent to the assertion that the rank of \(N(H)\) is the number of nonprojective simple \(H\)-modules in characteristic \(p\). This makes more precise an earlier Mackey functor version of Alperin's weight conjecture by \textit{J. Thévenaz} and \textit{P. J. Webb} [Astérisque 181/182, 263-272 (1990; Zbl 0738.20004)]. Also the author's theory of generalized Steinberg modules can be shown to fit into the framework of this paper.
category of finitely generated \(R\)-modules
isomorphism class of finite \(G\)-sets-\(H\)
scientific article; zbMATH DE number 923704