On functorial decompositions of self-smash products

Abstract

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 ℝP2, ℂP2, ℍP2 or double-struck K sign 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 ℤ/2[Sn] 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 ℤ/2[S n] of the indecomposable projective modules which correspond to those partitions.