= Submodule lattice of the standard symmetric-group permutation module
{title2=$T=F\mathbf1,\quad U=\ker(\sum_i)$}
For $n\geq3$, let $S_n$ permute the coordinates of $M=F^n$. Write $T$ for the constant vectors and $U=S^{(n-1,1)}$ for the augmentation <submodule>. These and zero and $M$ are all the <submodules>. Indeed a vector with two unequal coordinates yields a nonzero multiple of $e_i-e_j$ by subtracting its <image> under $(i,j)$; the <orbit> of such a difference spans $U$. A <submodule> without such a vector lies in $T$. If the <characteristic> does not divide $n$, then $M=T\oplus U$ and $U$ is simple. If it divides $n$, the lattice is the chain $0<T<U<M$, with $U/T$ simple of <dimension> $n-2$ and trivial end factors. The direct-sum <projection> in the first case is $x\mapsto x-(\sum_i x_i)\mathbf1/n$.
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