On functorial equivalence classes of blocks of finite groups

Abstract

Let k be an algebraically closed field of characteristic $p>0$ and let $F$ be an algebraically closed field of characteristic 0. Recently, together with Bouc, we introduced the notion of functorial equivalences between blocks of finite groups and proved that given a p-group D, there is only a finite number of pairs $(G,b)$ of a finite group G and a block b of kG with defect groups isomorphic to D, up to functorial equivalence over $F$. In this paper, we classify the functorial equivalence classes over $F$ of blocks with cyclic defect groups and 2-blocks of defects 2 and 3. In particular, we prove that for all these blocks, the functorial equivalence classes depend only on the fusion system of the block.