I am writing a paper(physics) where I am using the fact that the **irreducible's of the regular representations** of the **permutation group** are **absolutely irreducible** in the following sense.


If $V$ is an irreducible of the regular representation of $S\_n$ over $R$ (real numbers) then it remain irreducible under the extension of the field to complex numbers $ C$.


In particular,I only need the fact that they remain irreducible when we go from $R$ to $C$. 


Q1) Firstly , is this TRUE?


Q2) Can anyone please provide a reference where this is explicitly mentioned? (if possible also where in the text it appears)



Reference Request: Irreducibles of the regular representation of the permutation group is absolutely irreducible
Actually, all of these representations are defined over $\mathbb{Q}$ (i.e., they are absolutely indecomposable). This is stated in the second sentence of the Encyclopedia of Math article on [Representation of the symmetric groups](https://www.encyclopediaofmath.org/index.php/Representation_of_the_symmetric_groups). This follows immediately from the fact that the Young symmetrizer is defined over $\mathbb{Q}$ (and thus also $\mathbb{R}$).


If you have a simple representation $V$ of an arbitrary group over the reals, and you extend it to the complexes, it might not stay irreducible. But if that is the case, then the smaller representations can't be defined over the reals. If we have an isomorphism $\mathbb{C}\otimes\_{\mathbb{R}}V\cong V\_1\oplus V\_2$, then we must have that $V$ is isomorphic to $V\_1$, thought of as a real vector space of twice the dimension. If $V\_1$ had a real form, then thinking of it as a real vector space would give two copies of the same real representation, not an irreducible.


