On stable equivalences and blocks with one simple module. (Q961018)

Let \(p\) be a prime, and let \(B\) be a \(p\)-block of a finite group \(G\) with a defect group \(D\) of order \(|D|\leq p^2\). Suppose that the Brauer correspondent \(C\) of \(B\) has precisely one irreducible Brauer character. The authors show that there is a \(p\)-permutation equivalence between \(B\) and \(C\) inducing an isotypy. In particular, \(B\) has also precisely one irreducible Brauer character, and the decomposition numbers of \(B\) and \(C\) coincide. -- One of the main tools in their proof is a stable equivalence constructed by Rouquier.