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.Littelmannsoperatorsformthebasisofhiselegantandpowerfulpath model of the representation theory of classical groups. In our Appendix we use Littelmanns 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.