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

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 F Delta Rppk of diagonal p-permutation functors over R. We first show that the category FF Delta ppk 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 RT Delta G,b in F Delta Rppk. We find the decomposition of the functor FT Delta G,b as a direct sum of simple functors in FF Delta ppk. This leads to a characterization of nilpotent blocks in terms of their associated functors in FF Delta ppk.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.(c) 2022 Elsevier Inc. All rights reserved.