Cluster algebra structure on the finite dimensional representations of $U_q(\widehat{A_3})$ for $l$=2

In this paper, we prove one case of the conjecture given by Hernandez and Leclerc\cite{HL0}. Specifically, we give a cluster algebra structure on the Grothendieck ring of a full subcategory of the finite dimensional representations of a simply-laced quantum affine algebra $U_q(\widehat{\text{g}})$. In the procedure, we also give a specific description of compatible subsets of type $E_6$. As a conclusion, for every exchange relation of cluster algebra there exists a exact sequence of the full subcategory corresponding to it.