The Yang-Baxter equation (YBE) lies at the heart of many subjects, including quantum statistical mechanics, QFT, knot theory, braid groups, subfactors, quantum groups, quantum information ... . In this talk, I will consider involutive solutions of the YBE ("R-matrices"). Any such R-matrix defines a representation and an extremal character of the infinite symmetric group as well as a corresponding tower of subfactors. Using these structures, I will describe how to find all R-matrices up to a natural notion of equivalence inherited from applications in QFT (given by the character and the dimension), how to completely parameterize the set of solutions, and how to decide efficiently whether two given R-matrices are equivalent.