Intertwiners of representations of untwisted quantum affine algebras and Yangians revisited

Abstract: We discuss applications of the $q$-characters to the computation of the $R$-matrices. In particular, we describe the $R$-matrix acting in the tensor square of the first fundamental representation of E$_8$ and in a number of other cases, where the decomposition of the tensor squares with respect to non-affine quantum algebra has non-trivial multiplicities. As an illustration, we also recover $R$-matrices acting in the multiplicity free-case on the tensor squares of the first fundamental representations of all other types of untwisted quantum affine algebras. The answer is written in terms of projectors related to the decomposition of the tensor squares with respect to non-affine quantum algebras. Then we give explicit expressions for the $R$-matrices in terms of matrix units with respect to a natural basis (except for the case of E$_8$). We give similar formulas for the Yangian $R$-matrices.