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What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$? I ...

The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...

The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. ...