We explain various properties and applications of the skew RSK dynamics, which we introduced recently as a time evolution for a pair of skew Young tableaux (P,Q) [1]. The dynamics exhibits solitonic behaviors similar to box ball systems (BBS). Associated affine crystal structure allows to give a bijective proof of the Cauchy identity for the q-Whittaker polynomials. Its refinement provides a connection between Kardar-Parisi-Zhang(KPZ) models and free fermions at finite temperature.
[1] T. Imamura, M. Mucciconi, T. Sasamoto, Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials, arXiv: 2106.11922.