Q:

Doubly transitive group has irreducible augmentation module follow up

This is a follow-up question from Doubly transitive group has irreducible augmentation module?
If $\mathbb{k}$ is of characteristic zero, it is well known that $W$ is irreducible. How can one prove it?

A:

Let $H$ be a point stabilizer and $\pi=1_H\uparrow^G$ be the permutation character for the natural permutation action. Then (Frobenius reciprocity)
$$
(\pi,\pi)_G=(\pi,1_H\uparrow^G)_G=(\pi|H,1_H)=2
$$
since $H$ has two orbits. But then $\pi=1+\chi$ for an irreducible character $\chi$. OTOH $\chi=\pi-1$ is the action on $W$.

