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arXiv · 1402.6261

Electroid varieties and a compactification of the space of electrical networks

Thomas Lam
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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.

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Thomas Lam. 2014-02-25. Electroid varieties and a compactification of the space of electrical networks. https://arxiv.org/abs/1402.6261

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