Growth Diagrams and Non-symmetric Cauchy Identities on NW (SE) Near Staircases

The Robinson-Schensted-Knuth (RSK) correspondence is an important combinatorial bijection between two line arrays of positive integers (or non-negative integer matrices) and pairs of semi-standard Young tableaux (SSYTs). One of its applications, in the theory of Schur polynomials, is a bijective proof of the well known Cauchy identity. An interesting analogue of this bijection was given by Mason, where SSYTs are replaced by semi-skyline augmented fillings (SSAFs), originated in the Haglund-Haiman-Loehr formula for non-symmetric Macdonald polynomials. The latter object SSAF has the advantage of detecting the key of a SSYT which is easily read off from the SSAF shape. Using this analogue, we have previously considered the restriction of RSK correspondence to multisets of cells in a (truncated) staircase. The image is described by a Bruhat order inequality between the keys of the recording and the insertion fillings. This has allowed to derive a (truncated) triangular version of the Cauchy identity, due to Lascoux, where Schur polynomials are replaced by key polynomials or Demazure characters. We now consider the restriction of RSK to a near staircase, in French convention, where the top leftmost and the bottom rightmost cells and also possibly some cells in the diagonal layer are deleted. The image is described by additional Bruhat order inequalities, specified by the cells in the diagonal layer. The bijection is then used to extend the triangular version to near staircases, also a version due to Lascoux, where Demazure characters are now under the action of Demazure operators specified by the cells in the diagonal layer. Our analysis is made in the framework of Fomin’s growth diagrams where a formulation of the Mason’s analogue is given. This is then used to show how to pass from a triangular shape to a near staircase, via the action crystal operators, and how this affects the keys in the image of the RSK.