  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 $P(\lambda)$ of the simple module
$D(\lambda)$ where $\lambda$ 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.
