The globally irreducible representations of symmetric groups

Abstract:

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 $\mathcal {O} G$-module. We say that $M$ is a globally irreducible $\mathcal {O} G$-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 $\Sigma _n$.