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The Irreducible Representations of the Symmetric Groups

The Irreducible Representations of the Symmetric Groups THE IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRIC GROUPS G. D. JAMES 1. Introduction The purpose of this paper is to give an easy construction for all the irreducible representations of the symmetric groups over an arbitrary field. For each Young diagram D, of row lengths r ..., r the permutation module of S on the subgroup u ki n S x ... x S is denoted by V . A bilinear form is put on V then a submodule n rk D Di E of V is defined. E is, in fact, isomorphic to the Specht module [3] associated D D D with D. It turns out that as D varies (E /E n ED ) gives all the irreducible repre- D D sentations of S . When the field over which the representations are defined is of characteristic p, it seems to be difficult to find all the irreducible constituents of E n E . Examples D D of the methods which can be employed appear in the author's paper [1]. 2. Notation Throughout this paper n will be a fixed positive integer, and S will be the symmetric group on n letters. A Young diagram D is defined http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

The Irreducible Representations of the Symmetric Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/8.3.229
Publisher site
See Article on Publisher Site

Abstract

THE IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRIC GROUPS G. D. JAMES 1. Introduction The purpose of this paper is to give an easy construction for all the irreducible representations of the symmetric groups over an arbitrary field. For each Young diagram D, of row lengths r ..., r the permutation module of S on the subgroup u ki n S x ... x S is denoted by V . A bilinear form is put on V then a submodule n rk D Di E of V is defined. E is, in fact, isomorphic to the Specht module [3] associated D D D with D. It turns out that as D varies (E /E n ED ) gives all the irreducible repre- D D sentations of S . When the field over which the representations are defined is of characteristic p, it seems to be difficult to find all the irreducible constituents of E n E . Examples D D of the methods which can be employed appear in the author's paper [1]. 2. Notation Throughout this paper n will be a fixed positive integer, and S will be the symmetric group on n letters. A Young diagram D is defined

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1976

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