On functorial equivalence classes of blocks of finite groups

TL;DR

This paper classifies functorial equivalence classes of blocks of finite groups with specific defect groups, showing they depend only on the fusion system, thus advancing understanding of block classification in modular representation theory.

Contribution

It provides a classification of functorial equivalence classes for blocks with cyclic and small defect groups, linking these classes solely to their fusion systems.

Findings

Functorial equivalence classes depend only on the fusion system.

Complete classification for blocks with cyclic defect groups.

Results extend to 2-blocks with defects 2 and 3.

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.