Integrable models through representations of the hecke algebra

We present new solutions to the Yang-Baxter equation through representations of the Hecke algebra. The generators of the Hecke algebra are considered as Boltzmann weights for face models. The heights live in a graph. These models were conjectured to be integrable by Di Francesco and Zuber. We prove integrability for some of the suggested models by building explicitly the Boltzmann weights. Some relations that the Boltzmann weights satisfy and the consequence on partition functions with various boundary conditions are also discussed.