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Polynomial Representations of GL_n

with an Appendix on Schensted Correspondence and Littelmann Paths

  • Book
  • © 2007

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Authors:
  1. James A. Green
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  2. Manfred Schocker

    1. Department of Mathematics, University of Wales Swansea, Singleton Park, UK

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  3. Karin Erdmann

    1. Mathematical Institute, University of Oxford, OX1 3LB, UK

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Part of the book series: Lecture Notes in Mathematics (LNM, volume 830)

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Table of contents (6 chapters)

  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 Dλ,K

    Pages 33-42

  6. The Carter-Lusztig modules Vλ,K

    Pages 43-52

  7. Representation theory of the symmetric group

    Pages 53-70

  8. Back Matter

    Pages 72-163
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About this book

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.

Reviews

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)."

Authors and Affiliations

  • 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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