The problem of determining the elements of decomposition matrices of symmetric groups (Q788827)

Let \(D_ n\) be the matrix of decomposition numbers of the symmetric group of degree n over a field of characteristic 2, i.e. the matrix with rows indexed by partitions \(\lambda\) of n and columns indexed by 2- modular representations A of \(S_ n\), in which the (\(\lambda\),A)-entry is equal to the number of composition factors of the Specht module corresponding to \(\lambda\) which are isomorphic to A. The rows of \(D_ n\) for \(\lambda =(n-m,m)\) and \(\lambda =(n-r-1,r,1)\) are known [\textit{G. D. James}, The representation theory of the symmetric groups, Lect. Notes Math. 682 (1978; Zbl 0393.20009)]. The author determines the rows of \(D_ n\) for \(\lambda =(n-m,1,...,1)\).