Nakajima 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 thesis, 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.
There are 2 main parts to our work. The bulk of the work is a combinatorial construction that proves (q, t)-characters of a certain set of Kirillov-Reshetikhin modules t-commute under Nakajima’s twisted multi- plication. We use a slightly modified version of the tableaux-sum notation for q-characters and define the notion of a block-tableau, which plays an integral role in the proof. Once t-commutativity is established, the second half of this thesis is concerned with the commutation coefficients of the given set of Kirillov-Reshetikhin modules. In particular, we show that the commutation coefficients are compatible with the cluster algebra exchange matrix and the mutation relations in the language of Berenstein-Zelevinsky
