NW-SE expansions of non-symmetric Cauchy kernels on near staircases and growth diagrams

TL;DR

This paper provides a bijective proof for expansions of non-symmetric Cauchy kernels on near staircases, explicitly characterizing tableau pairs using growth diagrams and extending Mason's RSK analogue.

Contribution

It introduces an explicit bijective proof for near staircase expansions, clarifying tableau-pair characterizations via growth diagrams and semi-skyline augmented fillings.

Findings

Explicit tableau-pair characterization for near staircase expansions

Growth diagram formulation of Mason's RSK analogue

Extension of Lascoux's staircase expansion to near staircases

Abstract

Lascoux has given a triangular version of the Cauchy identity where Schur polynomials are replaced by Demazure characters and Demazure atoms. He has then used the staircase expansion to recover expansions for all Ferrers shapes, where the Demazure characters and Demazure atoms are under the action of Demazure operators specified by the cells above the staircase. The characterisation of the tableau-pairs in these last expansions is less explicit. We give here a bijective proof for expansions over near staircases, where the tableau-pairs are made explicit. Our analysis formulates Mason's RSK analogue, for semi-skylines augmented fillings, in terms of growth diagrams.