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.



Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$
Irreducible representations of $S\_n$ are *absolutely irreducible*, meaning that they remain irreducible after extension of scalars. Therefore, if $V$ is irreducible as an $\mathbb{R} S\_n$-module and $\phi:V\longrightarrow V$ is any nonzero homomorphism, then
$$\phi\otimes 1:V\otimes\mathbb{C}\longrightarrow V\otimes\mathbb{C}$$
is nonzero and $V\otimes \mathbb{C}$ is irreducible. It follows that $\phi\otimes1$ is an isomorphism, hence $\phi$ is an isomorphism.


To read more, I suggest Gordon James' book, or the James-Kerber text.


