= Radical of the augmentation Specht module
{title2=$\operatorname{rad}S^{(n-1,1)}=\begin{cases}F(1,\ldots,1),&p\mid n,\\0,&p\nmid n.\end{cases}$}
The <Specht module> $S^{(n-1,1)}$ is the augmentation subspace of the natural permutation module $F^n$, with the standard <tabloid bilinear form>. Its orthogonal complement in $F^n$ is the constant-vector line. That line lies in the augmentation subspace exactly when the <field> characteristic divides $n$, giving the displayed radical. Equivalently, the <basis> $v_i-v_n$ has <Gram matrix> $I_{n-1}+J_{n-1}$. The <James submodule theorem> makes every nonzero radical quotient simple, with <dimension> $n-2$ when $p\mid n$ and $n-1$ otherwise. The case $n=2,p=2$ gives the zero quotient rather than a nonzero <simple module>.
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