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