Polynomial Representations of GL_n

1. Front Matter

   Pages I-IX

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2. Introduction

   Pages 1-10

3. Polynomial Representations of GLn(K): The Schur algebra

   Pages 11-22

4. Weights and Characters

   Pages 23-31

5. The modules D_λ,K

   Pages 33-42

6. The Carter-Lusztig modules V_λ,K

   Pages 43-52

7. Representation theory of the symmetric group

   Pages 53-70

8. Back Matter

   Pages 72-163

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This second edition of “Polynomial representations of GL (K)” consists of n two parts. The ?rst part is a corrected version of the original text, formatted A in LT X, and retaining the original numbering of sections, equations, etc. E The second is an Appendix, which is largely independent of the ?rst part, but whichleadstoanalgebraL(n,r),de?nedbyP.Littelmann,whichisanalogous to the Schur algebra S(n,r). It is hoped that, in the future, there will be a structure theory of L(n,r) rather like that which underlies the construction of Kac-Moody Lie algebras. We use two operators which act on “words”. The ?rst of these is due to C. Schensted (1961). The second is due to Littelmann, and goes back to a1938paperbyG.deB.Robinsonontherepresentationsofa?nitesymmetric group.Littelmann’soperatorsformthebasisofhiselegantandpowerful“path model” of the representation theory of classical groups. In our Appendix we use Littelmann’s theory only in its simplest case, i.e. for GL . n Essential to my plan was to establish two basic facts connecting the op- ations of Schensted and Littelmann. To these “facts”, or rather conjectures, I gave the names Theorem A and Proposition B. Many examples suggested that these conjectures are true, and not particularly deep. But I could not prove either of them.

• Combinatorics
• Representation theory
• Schur algebra
• Young tableaux
• algebra
• combinatorics on words
• polynomial representation of the general linear group
• representation of the symmetric group

From the reviews: LNM 830 "is now regarded as the standard text on the finite-dimensional polynomial representations of the general linear group GL_n(K)."

• Department of Mathematics, University of Wales Swansea, Singleton Park, UK

  Manfred Schocker

• Mathematical Institute, University of Oxford, OX1 3LB, UK

  Karin Erdmann

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