Cluster realization of Weyl groups and $q$-characters of quantum affine algebras

We consider an infinite quiver $Q(\mathfrak{g})$ and a family of periodic quivers $Q_m(\mathfrak{g})$ for a finite dimensional simple Lie algebra $\mathfrak{g}$ and $m \in \mathbb{Z}_{>1}$. The quiver $Q(\mathfrak{g})$ is essentially same as what introduced by Hernandez and Leclerc for the quantum affine algebra. We construct the Weyl group $W(\mathfrak{g})$ as a subgroup of the cluster modular group for $Q_m(\mathfrak{g})$, in a similar way as what studied by the author, Ishibashi and Oya, and study its applications to the $q$-characters of quantum non-twisted affine algebras $U_q(\hat{\mathfrak{g}})$ introduced by Frenkel and Reshetikhin, and to the lattice $\mathfrak{g}$-Toda field theory. In particular, when $q$ is a root of unity, we prove that the $q$-character is invariant under the Weyl group action. We also show that the $A$-variables for $Q(\mathfrak{g})$ correspond to the $\tau$-function for the lattice $\mathfrak{g}$-Toda field equation.