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

We study the representation theory of the symmetric group $S_n$ in positive characteristic $p$. Using features of the LLT-algorithm we give a conjectural description of the projective cover $P(λ)$ of the simple module $D(λ)$ 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 Khovanov-Lauda-Rouquier algebras we explain an algorithm that allows us to verify this conjectural description for $n \leq 15$, at least.