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 $$Q_m({\mathfrak for finite-dimensional simple Lie algebra $${\mathfrak {g}}$$ $$m \in {\mathbb Z}_{>1}$$ . The is essentially same as what introduced in Hernandez Leclerc (J Eur Math Soc 18:1113–1159, 2016) the quantum affine $${\hat{{\mathfrak {g}}}}$$ construct Weyl group $$W({\mathfrak subgroup cluster modular , similar way (Inoue et al. Cluster realizations groups higher Teichmüller theory. arXiv:1902.02716 ), study its applications to q-characters non-twisted algebras $$U_q({\hat{{\mathfrak {g}}}})$$ (Frenkel Reshetikhin Contemp 248:163–205, 1999), lattice -Toda field theory Hikami Nucl Phys B 581:761–775, 2000). In particular, when q root unity, we prove that q-character invariant under action. also show A-variables correspond $$\tau $$ -function equation.