电网络空间的电簇与紧化
Electroid varieties and a compactification of the space of electrical networks
摘要：
本文利用仙人掌网络构造了Curtis-Ingerman-Morrow和De Verdiere-Gitler-Vertigan研究的圆形平面电网络空间的紧化。我们嵌入这个紧化作为一个线性切片的完全非负格拉斯曼，并与凯尼恩和威尔逊的格罗夫测量Postnikov的边界测量。切片与正向分层的交叉导致一类由匹配索引的电子变体。本文证明了电网络中匹配的偏序与仿射Bruhat序子集是对偶的。在这种情况下，拟阵的类似物是某些非交叉划分的特殊集合。
英文摘要：
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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