The Skeleton of a Reduced Word and a Correspondence of Edelman and Greene

Abstract

Stanley conjectured that the number of maximal chains in the weak Bruhat order of S n , or equivalently the number of reduced decompositions of the reverse of the identity permutation revn = n, n − 1, n − 2,..., 2, 1, equals the number of standard Young tableaux of staircase shape s = {n − 1, n −2,...,1}. Originating from this conjecture remarkable connections between standard Young tableaux and reduced words have been discovered. Stanley proved his conjecture algebraicaly, later Edelman and Greene found a bijective proof. We provide a generalization of the Edelman and Greene bijection to a larger class of words. The proof is inspired by Viennot’s planarized proof of the Robinson-Schensted correspondence. As it is the case with the classical correspondence the planarized proofs have their own beauty and simplicity.KeywordsYoung TableauMaximal ChainCover GraphWiring DiagramStandard Young TableauThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.