Soliton cellular automata associated with infinite reduced words

We consider a family of cellular automata $\mathrm{\Phi }(n,k)$ associated with infinite reduced elements on the affine symmetric group ${\widehat{S}}_n$, which is a tropicalization of the rational maps introduced by two of the authors. We study the soliton solutions for $\mathrm{\Phi }(n,k)$ and explore a `duality’ with the ${\mathfrak{s}\mathfrak{l}}_n$-box-ball system.