Calculus of tableaux

Abstract

There are two fundamental operations on tableaux from which most of their combinatorial properties can be deduced: the Schensted “bumping” algorithm, and the Schutzenberger “sliding” algorithm. When repeated, the first leads to the Robinson–Schensted–Knuth correspondence, and the second to the “jeu de taquin.” They are in fact closely related, and either can be used to define a product on the set of tableaux, making them into an associative monoid. This product is the basis of our approach to the Littlewood–Richardson rule. In Chapter 1 we describe these notions and state some of the main facts about them. The proofs involve relations among words which are associated to tableaux, and are given in the following two chapters. Chapters 4 and 5 have the applications to the Robinson–Schensted–Knuth correspondence and the Littlewood–Richardson rule. See Appendix A for some of the many possible variations on these themes.