Communications in Mathematical Physics

, Volume 341, Issue 3, pp 821–884 | Cite as

Twists of Plücker Coordinates as Dimer Partition Functions

  • R. J. MarshEmail author
  • J. S. Scott


The homogeneous coordinate ring of the Grassmannian Gr k,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plücker coordinates. We introduce a twist map on Gr k,n , related to the Berenstein–Fomin–Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plücker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plücker coordinate. We also relate the twist map to a maximal green sequence.


Bipartite Graph Cluster Variable Cluster Algebra Boundary Vertex Black Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of MathematicsUniversity of LeedsLeedsUK
  2. 2.Departamento de MatemáticasUniversidad de los AndesBogotáColombia

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