The Skew Column RSK dynamics and the Box and Ball System

Title: The Skew Column RSK dynamics and the Box and Ball System Abstract: We introduce a two-dimensional discrete integrable system, the \emph{Skew Column RSK Dynamics}, which is a two dimensional extension of the classical Box and Ball System (BBS) of Takahashi and Satsuma. The evolution acts deterministically on particle configurations over a periodic planar lattice, with local moves governed by the Fomin growth rules associated with the Robinson–Schensted–Knuth algorithm under column insertion. We construct a linearization algorithm that generalizes the Kerov–Kirillov–Reshetikhin (KKR) bijection, mapping the nonlinear particle dynamics to a linear evolution. Such linearization is stated as a bijection between pairs of semi-standard Young tableaux of skew-shape $(P,Q)$ and quadruples $(H_1,H_2;\kappa,\nu)$, where $H_1,H_2$ are horizontally weak tableaux encoding conservation laws of the dynamics, $\kappa$ is a list of non-negative integers and $\nu$ is a partition. As a by-product, we obtain bijective proofs of summation identities for modified Hall–Littlewood polynomials.