= Irreducible augmentation criterion for a transitive group action
For a transitive finite $G$-set $X$ with at least two points, the <augmentation subrepresentation of a permutation representation> is <irreducible> if and only if the action is <two-transitive>. Indeed, writing the permutation character as $\pi=1+\sum_i m_i\chi_i$ gives
$$
\langle\pi,\pi\rangle_G=1+\sum_i m_i^2.
$$
The augmentation subrepresentation is irreducible exactly when this norm is two. By the <character norm of a permutation representation>, that means that the diagonal and the off-diagonal are the only two orbits on $X\times X$, which is exactly two-transitivity.
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