Dual equivalence with applications, including a conjecture of Proctor

AbstractWe make a systematic study of a new concept in the theory of jeu-de-taquin, which we call dual equivalence. Using this, we prove a conjecture of Proctor establishing a bijection between standard tableaux of ‘shifted staircase’ shape and reduced expressions for the longest element in the Coxeter group Bl. We also get a new and more illuminating proof of the analogous theorem, due to Greene and Edelman, for the Coxeter group Al, and arrive at yet one more theorem of a similar type. We explain some symmetric functions associated to reduced expressions by Stanley and prove his conjecture that one of these for Bl is the Schur function sλ for λ an l-by-l square. We classify shifted and unshifted shapes for which the total promotion operator has special properties; in one case this proves another conjecture of Stanley. We determine the previously unknown ‘dual Knuth relations’ for the shifted Schensted correspondence