On the cluster nature and quantization of geometric \(R\)-matrices (Q1731795)

Summary: 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 \(U_{q'}(\hat {\mathfrak {sl}}_n)\) 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.