Growth Diagrams and Non-symmetric Cauchy Identities on Nw or Se near Staircases

Mason has introduced an analogue of the Robinson-Schensted-Knuth (RSK) correspondence to produce a bijection between biwords and pairs of semiskyline augmented fillings whose shapes, compositions, are rearrangements of each other. That pair of shapes encode the right keys for the pair of semi-standard Young tableaux produced by the usual Robinson-Schensted-Knuth (RSK) correspondence. We have shown that this analogue of RSK restricted to multisets of cells in staircases or truncated staircases allows expansions of non-symmetric Cauchy kernels in the basis of Demazure characters of type A, and the basis of Demazure atoms. One considers now a near staircase, in French convention, where the top leftmost and the bottom rightmost boxes of a staircase are deleted and also possibly some boxes in the diagonal layer. The conditions imposed on the pairs of shapes for the semi-skyline augmented fillings are described by inequalities in the Bruhat order, w.r.t. the symmetric group. The bijection is then used to provide a combinatorial expansion of an expansion formula, due to A. Lascoux, of a non-symmetric Cauchy kernel, over near staircases, in the basis of Demazure characters of type A, and the basis of Demazure atoms, under the action of appropriate Demazure operators. The analysis is made in the framework of Fomin’s growth diagrams for Robinson-SchenstedKnuth correspondences. On one hand, one gives a formulation of the analogue of RSK, via reverse RSK, to obtain pairs of semi-skyline augmented fillings, and, on the other hand, an interpretation of the action of crystal operators on biwords whose biletters are cells on a Ferrers shape. This sheds light on the aforesaid expansion provided by A. Lascoux.