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

DOI:
https://doi.org/10.1090/S0002-9939-99-05418-0

Published electronically:
July 27, 1999

MathSciNet review:
1676332

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a * globally irreducible* -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .

Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .

**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 . 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

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:
https://doi.org/10.1090/S0002-9939-99-05418-0

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