Diagonal p-permutation functors, semisimplicity, and functorial equivalence of blocks

Abstract

Let k be an algebraically closed field of characteristic p>0, let R be a commutative ring, and let F be an algebraically closed field of characteristic 0. We consider the R-linear category FRppkΔ of diagonal p-permutation functors over R. We first show that the category FFppkΔ is semisimple, and we give a parametrization of its simple objects, together with a description of their evaluations.Next, to any pair (G,b) of a finite group G and a block idempotent b of kG, we associate a diagonal p-permutation functor RTG,bΔ in FRppkΔ. We find the decomposition of the functor FTG,bΔ as a direct sum of simple functors in FFppkΔ. This leads to a characterization of nilpotent blocks in terms of their associated functors in FFppkΔ.Finally, for such pairs (G,b) of a finite group and a block idempotent, we introduce the notion of functorial equivalence over R, which (in the case R=Z) is slightly weaker than p-permutation equivalence, and we prove a corresponding finiteness theorem: for a given finite p-group D, there is only a finite number of pairs (G,b), where G is a finite group and b a block idempotent of kG with defect isomorphic to D, up to functorial equivalence over F.