Some remarks on G-functors and the Brauer morphism. (English) Zbl 0628.20011
If G is a finite group and k a commutative ring, J. A. Green defines a G- functor A over k to be a family of k-algebras \(A(H)\), with H running over the set of subgroups of G, together with restriction maps \(r^ K_ H: A(K)\to A(H)\), transfer maps \(t^ K_ H: A(H)\to A(K)\) (with \(H\leq K)\) and conjugation maps \(A(H)\to A(gHg^{-1})\), such that certain axioms are satisfied (included Mackey’s formula and Frobenius reciprocity). The Brauer morphism \(br^ G_ H: A(G)\to \bar A(H)=A(H)/\sum_{S<H}t^ H_ S(A(S))\) is by definition the composition of the restriction \(r^ G_ H\) and the canonical map \(br^ H_ H: A(H)\to \bar A(H)\). The set \({\mathcal P}(A)\) of subgroups H of G such that \(\bar A(H)\neq 0\) is shown to be a refinement of the concept of defect base \({\mathcal D}(A)\) introduced by Green: \({\mathcal D}(A)\) is the subgroup closure of \({\mathcal P}(A).\)
The product of all Brauer morphisms \(\beta_ G: A(G)\to (\prod_{H}\bar A(H))^ G\) terminates in the G-fixed points of \(\Pi_ H\bar A(H)\). The main result of the paper says on one hand that both Ker\(\beta_ G\) and Coker \(\beta_ G\) are annihilated by a power of \(| G|\) and on the other hand that Ker\(\beta_ G\) is a nilpotent ideal of \(A(G)\). In particular \(\beta_ G\) is an isomorphism if \(| G|\) is invertible in k and this fact is exploited in various ways. In particular the universal role of the Burnside ring G-functor is used to derive an idempotent formula and an induction formula generalizing known results. Also the notion of symmetric G-functor is defined and analyzed. The character ring and the Green ring are two important examples which are discussed in the light of the results of the paper.
The product of all Brauer morphisms \(\beta_ G: A(G)\to (\prod_{H}\bar A(H))^ G\) terminates in the G-fixed points of \(\Pi_ H\bar A(H)\). The main result of the paper says on one hand that both Ker\(\beta_ G\) and Coker \(\beta_ G\) are annihilated by a power of \(| G|\) and on the other hand that Ker\(\beta_ G\) is a nilpotent ideal of \(A(G)\). In particular \(\beta_ G\) is an isomorphism if \(| G|\) is invertible in k and this fact is exploited in various ways. In particular the universal role of the Burnside ring G-functor is used to derive an idempotent formula and an induction formula generalizing known results. Also the notion of symmetric G-functor is defined and analyzed. The character ring and the Green ring are two important examples which are discussed in the light of the results of the paper.
MSC:
20C15 | Ordinary representations and characters |
20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
20D30 | Series and lattices of subgroups |