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Young modules for symmetric groups

Part of: Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Karin Erdmann
Karin Erdmann
Affiliation:
Mathematical Institute24-29 St. Giles, Oxford OX1 3LB, United Kingdom, e-mail: [email protected]
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Abstract

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Let K be a field of characteristic p. The permutation modules associated to partitions of n, usually denoted as Mλ, play a central role not only for symmetric groups but also for general linear groups, via Schur algebras. The indecomposable direct summands of these Mλ were parametrized by James; they are now known as Young modules; and Klyachko and Grabmeier developed a ‘Green correspondence’ for Young modules. The original parametrization used Schur algebras; and James remarked that he did not know a proof using only the representation theory of symmetric groups. We will give such proof, and we will at the same time also prove the correspondence result, by using only the Brauer construction, which is valid for arbitrary finite groups.

MSC classification

Secondary: 20C30: Representations of finite symmetric groups 20C20: Modular representations and characters
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 2 , October 2001 , pp. 201 - 210
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Broué, M., ‘On Scott modules and p-permutation modules: an approach through the Brauer morphism’, Proc. Amer. Math. Soc. 93 (1985), 401–408.CrossRefGoogle Scholar
[2]Grabmeier, J., ‘Unzerlegbare Moduln mit trivialer Youngquelle und Darstellungstheorie der Schuralgebra’, Bayreuth. Math. Schr. 20 (1985), 9–152.Google Scholar
[3]James, G. D., The representation theory of the symmetric group, Lecture Notes in Math. 682 (Springer, Berlin, 1978).Google Scholar
[4]James, G. D., ‘Trivial source modules for symmetric groups’, Arch. Math. 41 (1983), 294–300.CrossRefGoogle Scholar
[5]James, G. D. and Kerber, A., The representation theory of the symmetric group, Encyclopedia Math. 16 (Addison and Wesley, Reading, 1981).Google Scholar
[6]Klyachko, A. A., ‘Direct summands of permutation modules’, Selecta Math. Soviet. (1) 3 (1983/1984), 45–55.Google Scholar