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Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Shih-Wei Yang & Andrei Zelevinsky

Authors
  1. Shih-Wei Yang
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  2. Andrei Zelevinsky
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Correspondence to Shih-Wei Yang.

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Dedicated to Bertram Kostant on the occasion of his 80th birthday

Both authors were supported by A. Zelevinsky’s NSF (DMS) grant # 0500534; A. Zelevinsky was also supported by a Humboldt Research Award.

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Yang, SW., Zelevinsky, A. Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors. Transformation Groups 13, 855–895 (2008). https://doi.org/10.1007/s00031-008-9025-x

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  • DOI: https://doi.org/10.1007/s00031-008-9025-x

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