Polynomial Representations of GL_n

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.

The new corrected and expanded edition adds a special appendix on Schensted Correspondence and Littelmann Paths. This appendix can be read independently of the rest of the volume and is an account of the Littelmann path model for the case gln. The appendix also offers complete proofs of classical theorems of Schensted and Knuth.

Includes supplementary material: sn.pub/extras

Preface to the second edition.- J. A. Green: Polynomial representations of GLn: 1.Introduction.- 2.Polynomial representations of GL_n(K): The Schur algebra.- 3.Weights and characters.- 4.The module D_{lambda, K}.- 5.The Carter-Lusztig modules V_{lambda, K}.- 6.Representation theory of the symmetric group.- Appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker: A. Introduction.- B. The Schensted process.- C. Schensted and Littelmann.- D. Theorem A and some of its consequences.- E. Tables.- Index of Symbols.- References.- Index.