Defect \(3\) blocks of symmetric group algebras (Q1283673)

Let \(k\) be a field of characteristic \(p\geq 5\) and let \(S_n\) denote the symmetric group of degree \(n\geq 1\). In a previous work, the authors studied the principal block \(B_0(kS_{3p})\), which is the unique block of \(kS_{3p}\) of defect \(3\). In particular they showed that, in \(B_0(kS_{3p})\), all decomposition numbers are \(0\) or \(1\), there are no self-extensions of irreducibles and that all extensions between irreducibles are of dimension at most \(1\). In this paper, they show that these properties hold for all defect \(3\) blocks of symmetric group algebras. In fact, their main theorem is: Theorem 7.1. Let \(p\geq 5\). A block \(B\) of \(kS_n\) of defect \(3\) has the following properties: (1) All of the decompositions are \(0\) or \(1\). (2) \((D^\lambda_k,D^\lambda_k)^1_{kS_n}=0\) for all \(p\)-regular \(\lambda\). (3) \((D^\lambda_k,D^\mu_k)=0\) or \(1\) for all \(p\)-regular \(\lambda\) and \(\mu\).