Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors
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We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan–Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.
KeywordsWeyl Group Simple Root Finite Type Cluster Variable Initial Cluster
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