In this talk, I will introduce the notion of functorial equivalence of blocks of finite groups, developed in recent joint work with Deniz Yilmaz. For a commutative ring R, and a field k of characteristic p>0, we introduce the category of diagonal p-permutation functors over R. To a pair (G,b) of a finite group G and a block idempotent b of kG, we associate a diagonal p-permutation functor FG,b, and we say that two such pairs (G,b) and (H,c) are functorially equivalent over R if the functors FG,b and FH,c are isomorphic. We show that the category of diagonal p-permutation functors over an algebraically closed field of characteristic 0 is semisimple. We obtain a precise description of the simple functors, and explicit formulas for their multiplicities as summands of FG,b. It follows that functorial equivalence preserves the defect groups of blocks, their number of simple modules, and their number of ordinary irreducible characters. This also leads to characterizations of nilpotent blocks, and to a finiteness theorem in the spirit of Donovan’s finiteness conjecture.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
