Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory.
I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb S_n$). I see that for the proof of Schur's lemma about the isormphisms $\phi:V \to V $ being the identity map times a scalar, when $V$ is an irreducible, one needs alebraically closed field.
$\mathbb R$ is not such a field. Is this lemma still true in this particular case?
I would be grateful if anyone can answer without too much sophisticated maths: If possible of course.