Mirror Symmetric Circular Planar Electrical Networks and Type B Electrical Lie Theory
نویسنده
چکیده
We introduce mirror symmetric circular planar electrical networks as the mirror symmetric subset of circular planar electrical networks studied by Curtis-Ingerman-Morrow [CIM] and Colin de Verdière-Gitler-Vertigan [dVGV]. These mirror symmetric networks can be viewed as type B generalization of circular planar electrical networks. We show that most of the properties of circular planar electrical networks are well inherited by these mirror symmetric electrical networks. Inspired by Lam [Lam], the space of mirror symmetric circular planar electrical networks can be compactified using mirror symmetric cactus networks, which admit a stratification indexed by mirror symmetric matchings. The partial order on the mirror symmetric matchings emerging from mirror symmetric electrical networks is dual to a subposet of affine Bruhat order of type C. We conjecture that this partial order is the closure partial order of the stratification of mirror symmetric cactus networks.
منابع مشابه
Electrical networks, electrical Lie algebras and Lie groups of finite Dynkin type
Curtis-Ingerman-Morrow studied the space of circular planar electrical networks, and classified all possible response matrices for such networks. Lam and Pylyavskyy found a Lie group EL2n whose positive part (EL2n)≥0 naturally acts on the circular planar electrical network via some combinatorial description, where the action is inspired by the star-triangle transformation of the electrical netw...
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