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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The globally irreducible representations
of symmetric groups


Authors: Alexander Kleshchev and Alexander Premet
Journal: Proc. Amer. Math. Soc. 128 (2000), 647-655
MSC (1991): Primary 20C30, 20C10
Published electronically: July 27, 1999
MathSciNet review: 1676332
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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 $\Sigma _n$.


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Additional Information

Alexander Kleshchev
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: klesh@math.uoregon.edu

Alexander Premet
Affiliation: Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
Email: sashap@ma.man.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05418-0
PII: S 0002-9939(99)05418-0
Keywords: Symmetric group, Specht module
Received by editor(s): December 10, 1997
Received by editor(s) in revised form: April 15, 1998
Published electronically: July 27, 1999
Additional Notes: The authors thank G. Michler and A. Zalesskii who organized a conference on representations of finite groups in Bad-Honnef where this collaboration began, and the Volkswagen foundation for financial support. The first author was also supported by the NSF
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1999 American Mathematical Society