The globally irreducible representations of symmetric groups
Alexander Kleshchev and Alexander Premet

Let $K$ be an algebraic number field and $\mathcal O$ be the ring of
integers of $K$. Let $G$ be a finite group and $M$ be a finitely generated
torsion free $\mathcal OG$-module. We say that $M$ is a {\em globally 
irreducible\/} $\mathcal OG$-module if for every maximal ideal $\mathfrak p$
of $\mathcal O$, the $k_{\mathfrak p}G$-module 
$M \otimes_{\mathcal O}k_{\mathfrak p}$ is irreducible, where $k_{\mathfrak p}$
stands for the residue field $\mathcal O/\mathfrak p$.

Answering a question of Pham Huu Tiep, we prove that the symmetric group
$\Sigma_n$ does not have non-trivial globally irreducible modules. More
precisely we establish that if $M$ is a globally irreducible 
$\mathcal O\Sigma_n$-module, then $M$ is an $\mathcal O$-module of rank $1$
with the trivial or sign action of $\Sigma_n$.