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The globally irreducible representations of symmetric groups

Authors:	Alexander Kleshchev Alexander Premet

Institution:	Department of Mathematics, University of Oregon, Eugene, Oregon 97403 ; Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

Abstract:	Let be an algebraic number field and $\mathcal{O}$ be the ring of integers of . Let be a finite group and be a finitely generated torsion free $\mathcal{O} G$ -module. We say that 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 is a globally irreducible $\mathcal{O}\, \Sigma _n$ -module, then is an $\mathcal{O}$ -module of rank with the trivial or sign action of $\Sigma _n$ .

Keywords:	Symmetric group Specht module

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