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

We consider an infinite quiver Q(g) and a family of periodic quivers Q(m)(g) for a finitedimensional simple Lie algebra g and m is an element of Z(>1). The quiver Q(g) is essentially same as what introduced in Hernandez and Leclerc (J Eur Math Soc 18:1113-1159, 2016) for the quantum affine algebra (g) over cap. We construct the Weyl group W(g) as a subgroup of the cluster modular group for Q(m)(g), in a similar way as (Inoue et al. in Cluster realizations of Weyl groups and higher Teichmuller theory. arXiv:1902.02716), and study its applications to the q-characters of quantum non-twisted affine algebras U-q ((g) over cap) (Frenkel and Reshetikhin in Contemp Math 248:163-205, 1999), and to the lattice gToda field theory (Inoue and Hikami inNucl Phys B 581:761-775, 2000). 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(g) correspond to the t -function for the lattice g-Toda field equation.