井上 玲

We consider a family of cellular automata $\Phi(n,k)$ associated with<br /> infinite reduced elements on the affine symmetric group $\hat S_n$, which is a<br /> tropicalization of the rational maps introduced by two of the authors. We study<br /> the soliton solutions for $\Phi(n,k)$ and explore a `duality&#039; with the<br /> $\mathfrak{sl}_n$-box-ball system.

We define cluster $R$-matrices as sequences of mutations in triangular grid<br /> quivers on a cylinder, and show that the affine geometric $R$-matrix of<br /> symmetric power representations for the quantum affine algebra<br /> $U_q^\prime(\hat{\mathfrak{sl } }_n)$ can be obtained from our cluster<br /> $R$-matrix. A quantization of the affine geometric $R$-matrix is defined,<br /> compatible with the cluster structure. We construct invariants of the quantum<br /> affine geometric $R$-matrix as quantum loop symmetric functions.