Twists of Pl\"ucker coordinates as dimer partition functions
Abstract
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.
Cite
@article{arxiv.1309.6630,
title = {Twists of Pl\"ucker coordinates as dimer partition functions},
author = {Bethany Marsh and Jeanne Scott},
journal= {arXiv preprint arXiv:1309.6630},
year = {2020}
}
Comments
61 pages, 30 figure files. References added. New Section 11 on relationship with maximal green sequences. Simplification of arguments in Section 3. End of Section 6 moved into Section 7. Minor corrections and improvements. To appear in Communications in Mathematical Physics