Schur operators and Knuth correspondences

The paper presents a general combinatorial approach to the Schur functions and their modifications, respective generalized Cauchy identities, and bijective Knuthtype 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 [3]. This paper is a revised version of [5].