I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say
$$ S_{n_1} \times \cdots \times S_{n_p} $$
is isomorphic to a tensor product $\rho_1 \otimes \cdots \otimes \rho_p$ where $\rho_i$ is an irreducible representation of $S_{n_i}$.
I have an open mind about this, but I'm imagining doing it by finding references for these two claims:
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If $k$ is an algebraically closed field of characteristic zero, every irreducible representation of a product $G_1 \times G_2$ of finite groups is of the form $\rho_1 \otimes \rho_2$ where $\rho_i$ is an irreducible representation of $G_i$.
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If $k$ has characteristic zero and $\overline{k}$ is its algebraic closure, every finite-dimensional representation of $S_n$ over $\overline{k}$ is isomorphic to one of the form $\overline{k} \otimes_k \rho$ where $\rho$ is a representation of $S_n$ over $k$.
Serre's book Linear Representations of Finite Groups states the first fact for $k = \mathbb{C}$ but apparently not for a general algebraically closed field of characteristic zero. (It's Theorem 10.) It could be true already for any field of characteristic zero, which would simplify my life.
The second fact should be equivalent to saying that $\mathbb{Q}$ is a splitting field for any symmetric group, which seems to be something everyone knows – yet I haven't found a good reference.
Best Answer
Question 1 is a special case of the following statement for finite dimensional algebras: Let $A$ and $B$ be a finite dimensional algebras over a field $K$ such that $A/rad(A)$ and $B/rad(B)$ are isomorphic to a direct product of matrix algebras over $K$ (which is always true when $K$ is algebraically closed).
When $e_i$ and $e_i'$ are pariwise orthogonal primitive idempotents which sum to 1 for $A$ and $B$ respectively, then $e_i \otimes_K e_i'$ are pairwise orthogonal primitive idempotents which sum to 1 for $A \otimes_K B$. Thus when $S_i$ and $S_i'$ are the simple $A$ and $B$-modules respectively, then $S_i \otimes S_i'$ are the simple $A \otimes_K B$ modules. This is proved for $A=B$ in the book "Frobenius algebras I" by Skowronski and Yamagata in chapter IV. as proposition 11.3. , but the proof works in exactly the same way when $A \neq B$.
For question 2, one can find in section 4.5. corollary 4.16 in the book "A tour of repreesntation theory" by Martin Lorenz the fact that $\mathbb{Q}$ is a splitting field for the symmetric group. The whole section 4 in this book is dedicated to the representation theory of the symmetric group in characteristic 0 and might be one of the nicest modern approaches to this problem.
Thus since $\mathbb{Q}$ is a splitting field for the symmetric group, it is true for any field $K$ of characteristic 0 (not just algebraically closed fields) that the irreducible representations of a direct product of symmetric groups is given as a tensor product of the irreducible representations of the single symmetric groups.