On the cluster nature and quantization of geometric $R$-matrices

Abstract

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