By serge Bouc
This quantity exposes the idea of biset functors for finite teams, which yields a unified framework for operations of induction, restrict, inflation, deflation and shipping through isomorphism. the 1st half remembers the fundamentals on biset different types and biset functors. the second one half is anxious with the Burnside functor and the functor of complicated characters, including semisimplicity matters and an outline of eco-friendly biset functors. The final half is dedicated to biset functors outlined over p-groups for a set major quantity p. This comprises the constitution of the functor of rational representations and rational p-biset functors. The final chapters disclose 3 functions of biset functors to long-standing open difficulties, specifically the constitution of the Dade workforce of an arbitrary finite p-group.This ebook is meant either to scholars and researchers, because it offers a didactic exposition of the fundamentals and a rewriting of complicated leads to the realm, with a few new rules and proofs.
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Extra info for Biset functors for finite groups
2. If U is a (G, G)-biset, and V is an (H, H)-biset, then U × V is a (G × H, G × H)-biset for the structure given by (g, h) · (u, v) · (g , h ) = (gug , huh ) , for g, g ∈ G, h, h ∈ H, u ∈ U , v ∈ V . The correspondence (U, V ) → U × V induces a bilinear map from B(G, G) × B(H, H) to B(G × H, G × H), hence a linear map π2 : B(G, G) ⊗Z B(H, H) → B(G × H, G × H) , which is an injective ring homomorphism, preserving identity elements. If G and H have coprime orders, this map is an isomorphism. Proof: In Assertion 1, the correspondence (X, Y ) → X × Y induces an obviously bilinear map from B(G) × B(H) to B(G × H), hence a linear map π from B(G) ⊗Z B(H) to B(G × H).
E. it is deﬁned by LG,V (ϕ)(f ⊗ v) = (ϕf ) ⊗ v , for f ∈ HomRD (G, H) and v ∈ V . Similarly, the right adjoint r IndD of the functor EvG maps the D EndRD (G)-module V to the biset functor LoG,V on D deﬁned by LoG,V (H) = HomEndRD (G) HomRD (H, G), V , where HomRD (H, G) is a left EndRD (G)-module by composition. When ϕ : H → H is a morphism in RD, the map LoG,V (ϕ) : LoG,V (H) → LoG,V (H ) is deﬁned by LoG,V (ϕ)(θ)(f ) = θ(f ϕ) , for θ ∈ LoG,V (H) and f ∈ HomRD (H , G). 4, there are isomorphisms of EndRD (G)-modules V ∼ = LG,V (G) and V ∼ = LoG,V (G).
The functor r IndD D ) is left adjoint (resp. right adjoint) to the functor ResD . 4. Proposition : Let D be a subcategory of C, containing group isomorphisms, and let D be a full subcategory of D. Then the functors D l ResD D ◦ IndD and D r ResD D ◦ IndD are isomorphic to the identity functor of FD ,R . D D l D r Proof: Since ResD D ◦ IndD is left adjoint to ResD ◦ IndD , it is enough to prove result for the latter. e. to the Yoneda functor Yd on the category RD . By the Yoneda Lemma, for any object F of FD ,R , the set of natural transformations from Yd to F is isomorphic to F (d ), and this isomorphism is D r functorial in d .
Biset functors for finite groups by serge Bouc