Schur Operators and Knuth Correspondences

The paper presents a general combinatorial approach to the Schur functions and their modiications, respective generalized Cauchy identities, and bijective Knuth-type correspondences between matrices and pairs of tableaux. All of these appear whenever one has a pair of graphs with the same vertices such that the linear operators associated with these graphs satisfy a certain type of commutation relations. A parallel implementation of insertion-type algorithms is suggested that generalizes the sequential constructions of Sagan and Stanley 13, 14] and the earlier bijections of Knuth, Worley-Sagan, and Haiman. We use the linear-algebraic approach of 17, 2] and the algorithmic techniques of 3]. This paper is a revised version of 5].