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?

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 $W$.