Projective modules for the symmetric group and Young's seminormal form

We study the representation theory of the symmetric group SnSn in positive characteristic p  . Using features of the LLT-algorithm we give a conjectural description of the projective cover P(λ)P(λ) of the simple module D(λ)D(λ) where λ is a p-restricted partition such that all ladders of the corresponding ladder partition are of order less than p  . Inspired by the recent theory of KLR-algebras, we explain an algorithm that allows us to verify this conjectural description for n≤15n≤15, at least.