Soliton cellular automata associated with infinite reduced words

Abstract

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 in [3]. We study the soliton solutions for $\Phi (n,k)$ and explore a duality with the ${\mathfrak{sl}}_n$-box-ball system.