Abstract:
This Appendix is devoted to the study of several combinatorial constructions involving standard Young tableaux (SYT) that lead to the proof of the Littlewood-Richardson rule, a combinatorial rule describing the coefficients in the Schur function expansion of an arbitrary skew Schur function (or in a product of two ordinary Schur functions). Most of what follows can be straightforwardly generalized to semistandard Young tableaux (SSYT). We do not do it here, in order to simplify the presentation. 1 Knuth equivalence and Greene's theorem The RSK correspondence w RSK \Gamma! (P; Q) associates to a permutation w 2 S n a pair of SYT: the insertion tableau P and the recording tableau Q; these tableaux have the same shape sh(w). In this section, we examine the following two questions: ffl What are the conditions for two permutations to have the same shape sh(w)? ffl What are the conditions for two permutations to have the same insertion tableaux P? The first question has an answer involving a particular family of poset-theoretic invariants of permutations. The equivalence relation appearing in the second question can be described in terms of certain elementary transformations that change three consecutive entries of a permutation. We first state these two results, and devote the rest of this section to their proof. For a permutation w = w 1 \Delta \Delta \Delta w n 2 S n and k 2 N, let I k = I k (w) denote the maximal number of elements in a union of k increasing subsequences of w. Analogously, let D l be the maximal size of a union of l decreasing subsequences of w. For example, for w = 236145 2 1
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