The Robinson-Schensted and Schützenberger algorithms, an elementary approach

We discuss the Robinson-Schensted and Schutzenberger algorithms, and the fundamental identities they satisfy, systematically interpreting Young tableaux as chains in the Young lattice. We also derive a Robinson-Schensted algorithm for the hyperoctahedral groups. Finally we show how the mentioned identities imply some fundamental properties of Schutzenberger's glissements. The two algorithms referred to in our title are combinatorial algorithms dealing with Young tableaux. The former was found originally by G. de B. Robinson (Rob), and independently rediscovered much later, and in a difierent form, by C. Schensted (Sche); it establishes a bijective correspondence between permutations and pairs of Young tableaux of equal shape. The latter algorithm (which is sometimes associated with the term jeu de taquin) was introduced by M. P. Schutzenberger (Schu1), who also demonstrated its great importance in relation to the former algorithm; it establishes a shape preserving involutory correspondence between Young tableaux. These algorithms have been studied mainly for their own sake|they exhibit quite remarkable combinatorial properties|rather than primarily serving (as is usually the case with algorithms) as a means of computing some mathematical value. 0.1. Some history. The Robinson-Schensted algorithm is the older of the two algorithms considered here. It was flrst de- scribed in 1938 by Robinson (Rob), in a paper dealing with the representation theory of the symmetric and general linear groups, and in particular with an attempt to prove the correctness of the rule that Littlewood and Richardson (LiRi) had given for the computation of the coe-cients in the decomposition of products of Schur functions. Robinson's description of the algorithm is rather obscure however, and his proof of the Littlewood-Richardson rule incomplete; apart from the fact that the supposed proof was reproduced in (Littl), the algorithm does not appear to have received much mention in the literature of the subsequent decades. The great interest the algorithm enjoys nowadays by combinatorialists was triggered by its independent reformulation by Schensted (Sche) published in 1961, whose main objective was counting permutations with given lengths of their longest increasing and decreasing subsequences; it was not recognised until several years later that this algorithm is essentially the same as Robinson's, despite its rather difierent deflnition. The combinatorial signiflcance of Schensted's algorithm was indi- cated by Schutzenberger (Schu1), who at the same time introduced the other algorithm that we shall be considering (the operation called I in (Schu1,x5)): he stated a number of important identities satisfled by the correspondences deflned by the two algorithms, and relations between them. That paper represents a big step forward in the understanding of the Robinson-Schensted algorithm, but the important results are somewhat obscured by the complicated notation and many minor errors, and by the fact that its emphasis lies on treating the limiting case of inflnite permutations and Young tableaux, a generalisation that has been ignored in the further development of the subject.

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