Radical of the augmentation Specht module ($\mathrm{rad}{S}^{(n-1,1)}=\{\begin{array}{ll}F(1,\dots ,1),& p\mid n,\\ 0,& p\nmid n.\end{array}$)

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.