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.

Kind regards.

irreducibleandabsolutely irreducible, as in David Hill's answer, is also discussed on Wikipedia (Schur's lemma). In this case, absolutely irreducible means irreducible even after complexification (extension of scalars from real to complex numbers). $\endgroup$ – Igor Khavkine Oct 22 '14 at 11:40