= Solution
Give $M^\lambda$ the invariant <symmetric bilinear form> for which its <tabloid> <basis> is orthonormal. Define
$$
\boxed{D^\lambda=S^\lambda/(S^\lambda\cap(S^\lambda)^\perp).}
$$
In characteristic $p>0$, this quotient is nonzero and simple when $\lambda$ is $p$-regular, meaning that no part occurs $p$ or more times. For other <integer partitions>[partitions] it is zero, so the label “simple <module>” includes the nonzero/$p$-regular qualification.
The <James submodule theorem> states that for every $F\Sigma_n$-<submodule> $U\subseteq M^\lambda$, either $S^\lambda\subseteq U$ or $U\subseteq(S^\lambda)^\perp$. Applied to <submodules> of $S^\lambda$, it says that every proper <submodule> lies in the radical of the restricted form, giving the simplicity of the displayed nonzero quotient.
Back to article page
