On the Cluster Nature and Quantization of Geometric $R$-Matrices

Source 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.