(q, t)-Characters of Kirillov-Reshetikhin Modules of Type Ar as Quantum Cluster Variables

Nakajima (2003) introduced a t-deformation of q-characters, (q, t)-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima (q, t)-characters of Kirillov-Reshetikhin modules satisfy a t-deformed T -system. The T -system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima t-deformed T system of type Ar forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the T -system.