A note on blocks of finite groups with TI Sylow \(p\)-subgroups (Q6130269)

In modular representation theory, there are different notions of equivalences between blocks of finite groups such as Puig equivalence, splendid Rickard equivalence, \(p\)-permutation equivalence, isotypies, and perfect isometries (see, e.g., [\textit{M. Broué}, Astérisque 181--182, 61--92 (1990; Zbl 0704.20010); \textit{R. Boltje} and \textit{B. Xu}, Trans. Am. Math. Soc. 360, No. 10, 5067--5087 (2008; Zbl 1175.20009)]). Each equivalence in that list implies the subsequent one. Let \(\mathbb{F}\) be an algebraically closed field of characteristic \(0\). In the paper under review the author proves that the functorial equivalence over \(\mathbb{F}\) and perfect isometry between blocks of finite groups do not imply each other.