On functorial equivalence classes of blocks of finite groups

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $\mathbb{F}$ 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 $\mathbb{F}$. In this paper, we classify the functorial equivalence classes over $\mathbb{F}$ 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.