Electroid varieties and a compactification of the space of electrical networks
Abstract
We construct a compactification of the space of circular planar electrical networks studied by Curtis-Ingerman-Morrow and De Verdiere-Gitler-Vertigan, using cactus networks. We embed this compactification as a linear slice of the totally nonnegative Grassmannian, and relate Kenyon and Wilson's grove measurements to Postnikov's boundary measurements. Intersections of the slice with the positroid stratification leads to a class of electroid varieties, indexed by matchings. The partial order on matchings arising from electrical networks is shown to be dual to a subposet of affine Bruhat order. The analogues of matroids in this setting are certain distinguished collections of non-crossing partitions.
Keywords
Cite
@article{arxiv.1402.6261,
title = {Electroid varieties and a compactification of the space of electrical networks},
author = {Thomas Lam},
journal= {arXiv preprint arXiv:1402.6261},
year = {2018}
}
Comments
44 pages. Version 2: minor corrections, references updated. Version 3: exposition improved, table of contents added. Version 4: minor corrections