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2018年10月26日(金) 17:30~18:30, 18:45~19:45
I-siteなんば 2F S1
大矢浩徳氏(芝浦工業大学)
Cluster realizations of Weyl groups and their application
Cluster algebras are commutative algebras which are determined from combinatorial data represented by weighted quivers. These combinatorial data are used for successive construction, called mutation, of algebra generators. A cluster modular group is the group consisting of automorphisms of a cluster algebra which are given by mutation sequences preserving a given weighted quiver. In this talk, we realize the Weyl groups associated with symmetrizable Kac-Moody Lie algebras as subgroups of cluster modular groups. We explain application of our construction of Weyl groups : An algebraic application is a systematic construction of green sequences associated with reduced words of elements of the Weyl group. In particular, if a given Kac-Moody Lie algebra is of finite type, then reduced words of the longest element give the maximal green sequences (and the cluster Donaldson-Thomas transformation) associated to our weighted quiver. As a geometric application, we discuss the comparison between our realization of the Weyl group and a geometric action of the Weyl group on the moduli space of twisted decorated $G$-local systems on a marked surface, which is known to be a cluster variety. This talk is based on a joint work with Rei Inoue and Tsukasa Ishibashi.