Cluster algebras and minimal affinizations of representations of the quantum group of type $G_2$

In this paper, we make a connection between cluster algebras and a class of modules of the quantum affine algebra $U_q \hat{\mathfrak{g}}$ of type $G_2$ called minimal affinizations. We introduce a system of equations satisfied by the $q$-characters of minimal affinizations of type $G_2$ which we called the M-system of type $G_2$. The M-system of type $G_2$ contains all minimal affinizations of type $G_2$ and only contains minimal affinizations. We show that the equations in the M-system of type $G_2$ correspond to mutations in some cluster algebra $\mathscr{A}$. Moreover, the minimal affinizations correspond to some cluster variables in $\mathscr{A}$.