2013/09/25 by Marsh, Bethany, Scott, Jeanne · 1 citation
#05E15 #14M15 #82B20 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Primary 13F60 #Representation Theory (math.RT) #Secondary 05C22 #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.1309.6630
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 Pluecker 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 Pluecker 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 Pluecker coordinate. We also relate the twist map to a maximal green sequence.