Q and T systems, paths and cluster positivity I

In the theory of lattice integrable models, one encounters a certain class of integrable discrete equations called Q-systems and T-systems. They describe the characters and q-characters of special quantum affine algebra modules. I will explain the relation of these equations to the positivity problem in cluster algebras and present a path model formulation for the solution in the case where the algebra is of type A. This is joint work with Philippe Di Francesco.