The skew RSK dynamics is a discrete 2 dimensional deterministic dynamics introduced by Imamura, Sasamoto and myself. It is an integrable system in the sense that it possesses infinitely many conservation laws and a rich set of symmetries. The analysis of its scattering relations produces a nontrivial correspondence which generalizes the celebrated Robinson-Schensted-Knuth bijection. Such correspondence provides a bijective proof of summation identities between special symmetric polynomials, which describe the law of certain interacting particle systems or directed polymer models and allow their asymptotic analysis.