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2020年12月4日(金) 15:00~16:30
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井上玲氏(千葉大学)
Cluster realization of Weyl groups and q-characters of quantum affine algebras
We introduce the Weyl group realization as a subgroup of the cluster modular group for some periodic quiver, and discuss its applications, based on arXiv: 2003.04491. We consider an infinite quiver Q(g) and a family of periodic quivers Q_m(g) for a finite dimensional simple Lie algebra g and an integer m bigger than one. The quiver Q(g) is essentially same as what introduced in [Hernandez-Leclerc 16] in studying the q-characters for quantum non-twisted affine algebras. For the quiver Q_m(g) we construct the Weyl group W(g) in a similar way as [I-Ishibashi-Oya 19], and study its applications to the q-characters [Frenkel-Reshetikhin 99], and to the lattice g-Toda field theory [I-Hikami 00]. 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 tau-functions for the lattice g-Toda field equation.