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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 $\Sigma _n$.

References [Enhancements On Off] (What's this?)

  • 1. R. W. Carter and M. T. J. Payne, On homomorphisms between Weyl modules and Specht modules, Math. Proc. Camb. Phil. Soc. 87 (1980), 419-425. MR 81h:20048
  • 2. C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, John Wiley and Sons (Interscience), New York, 1962. MR 26:2519
  • 3. W. and F. Ellison, Prime numbers, John Wiley and Sons, Hermann, 1985. MR 87a:11082
  • 4. G. D. James, On the decomposition matrices of the symmetric groups, II, J. Algebra 43 (1976), 45-54. MR 55:3057b
  • 5. G. D. James, On a conjecture of Carter concerning irreducible Specht modules. Math. Proc. Camb. Phil. Soc. 83 (1978), 11-17. MR 57:3234
  • 6. G. D. James, The representation theory of the symmetric groups, Springer Lecture Notes 682, Berlin, Heidelberg, New York, 1978. MR 80g:20019
  • 7. G. D. James and G. E. Murphy, The determinant of the Gram matrix for a Specht module. J. Algebra 59 (1979), 222-235. MR 82j:20025
  • 8. P. Ribbenboim, The book of prime number records, Springer-Verlag, New York, Berlin, Heidelberg, 1988.
  • 9. J. G. Thompson, Finite groups and even lattices. J. Algebra 38 (1976), 523-524. MR 53:3108
  • 10. J. G. Thompson, A simple subgroup of $E_{8}(3)$. In: N. Iwahori (ed.), Finite groups symposium, Japan Soc. for promotion of Science, 1976, pp. 113-116.
  • 11. Pham Huu Tiep, Globally irreducible representations of finite groups and integral lattices, Geometriae Dedicata 64 (1997), 85-123. MR 98e:20011

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Additional Information

Alexander Kleshchev
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Alexander Premet
Affiliation: Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

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

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