The globally irreducible representations of symmetric groups

Let K be an algebraic number field and O be the ring of integers of K. Let G be a finite group and M be a finitely generated torsion free OG-module. W e say that M is a globally irreducible OG-module if, for every maximal idea l p of O, the k(p) G-module M X (O)k(p) is irreducible, where k(p) stands f or the residue field O/p. Answering a question of Pham Huu Tiep, we prove that the symmetric group Si gma(n) does not have non-trivial globally irreducible modules. More precise ly we establish that if M is a globally irreducible O Sigma(n)-module, then M is an O-module of rank 1 with the trivial or sign action of Sigma(n).