On functorial decompositions of self-smash products

We give a decomposition formula for n-fold self smash of a two-cell suspension X localized at 2. The mod 2 homology of each factor in the decomposition is explicitly given as a module over the Steenrod algebra and in the case where X is formed by suspending one of P²,P²,P² or P², this is a complete decomposition into indecomposable pieces. The method has consequences in the modular representation theory of the symmetric group where it leads to a computation of the submatrix for the decomposition matrix of the group algebra /2[S_n] which correspond to partitions of length 2. In particular this yields a derivation of the explicit formula due to Erdmann which gives the multiplicities in the decomposition of /2[S_n] of the indecomposable projective modules which correspond to those partitions.