Decomposition numbers of symmetric groups by induction (Q1570870)

Let \(F\) be a field of characteristic \(p\), and let \(S_n\) denote the symmetric group on \(n\) letters. It is known that for each partition \(\lambda\) of \(n\) there is an \(FS_n\)-module \(S^\lambda\) called Specht module. The irreducible \(FS_n\)-module corresponding to a \(p\)-regular partition \(\mu\) of \(n\) is denoted by \(D^\mu\). The decomposition multiplicity of \(D^\mu\) in \(S^\lambda\) is called \(p\)-modular decomposition number of \(S_n\) and is denoted by \([S^\lambda:D^\mu]\). Finding these numbers is an important problem and there is no available method to evaluate them. But in the paper under review the authors try to find the \(p\)-modular decomposition numbers \([S^\lambda:D^\mu]\) of \(S_n\) in terms of \(p\)-modular decomposition numbers of a symmetric group with smaller degree. If \(\lambda\) has precisely three non-zero parts the authors prove a result which reduces finding \([S^\lambda:D^\mu]\) to a very special case.