Polynomial representations of \(\text{GL}_n\). With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker.
This second edition of Lect. Notes Math. 830 is twice the size of the old one [1980; Zbl 0451.20037]. The first half reproduces the classical treatment by Green of the polynomial representations of \(\text{GL}_n\). Here one learns about the Schur algebra, the Weyl modules, the Schur modules, the Specht modules. The treatment is purely algebraic and valid in any characteristic. One fixes a degree \(r\) for the polynomial representations. Then everything is done in terms of `words', where a word is a sequence of length \(r\) of integers between \(1\) and \(n\). The second half of the book is about fitting the Littelmann root operators into this picture. In the case of \(\text{GL}_n\) the paths of Littelmann's path model may be replaced with words. So that is done. Next one treats the Robinson-Schensted process and the Littelmann path operators. A key result, Proposition~B, is that the root operators commute with the process in an appropriate sense. The Robinson-Schensted process turns a word \(i\) into a triple \((\lambda(i),P(i),Q(i))\), where \(\lambda(i)\) is a partition, and \(P(i)\), \(Q(i)\) are tableaux. A theorem of Knuth describes a set of basic moves on words so that \(P(i)=P(j)\) if and only if the words \(i\) and \(j\) can be connected by a sequence of basic moves. Now Theorem~A tells that \(Q(i)=Q(j)\) if and only if \(i\) and \(j\) can be connected by applying a sequence of Littelmann root operators. One introduces the `Littelmann algebra' generated by the Littelmann root operators acting on the formal linear span of the set of words. It is seen to be analogous to the Schur algebra and its representations are described in terms of Theorem~A. At the end of the book the connection with the plactic monoid of Lascoux, Leclerc and Thibon is discussed.
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