AI 中文总结
本文研究平面电路网络的丛林计数与辛群不变理论的关系，构造辛不变张量，并在辛群情形下建立其与丛林公式及双丛林计数的联系。
AI 中文摘要
考虑一个具有正实边权和指定边界顶点（称为节点）的有限平面图。这样的图称为圆形平面电路网络。一个丛林是一个生成森林，其中每个连通分量至少包含一个节点。丛林的连通分量确定了节点的一个划分。我们将加权丛林计数与辛群的不变理论联系起来。对于一个平面电路网络 $G$，我们关联一个 $\mathrm{Sp}(2n)$-不变张量 $Z_G$。对于 $\mathrm{Sp}(2)=\mathrm{SL}(2)$，我们将 $Z_G$ 在以非交叉匹配为索引的 Temperley--Lieb 基中展开，并将其系数与 Kenyon--Wilson 丛林公式联系起来。对于 $\mathrm{Sp}(4)$，我们给出了两个丛林叠加的约化规则，并证明了以 $3$-非交叉匹配为索引的张量构成了 $\mathrm{Sp}(4)$-不变张量空间的一组基。在此基下，$Z_G$ 的系数是约化双丛林（reduced double groves）的加权计数，直至归一化。
英文摘要
Consider a finite planar graph with positive real edge weights and designated boundary vertices, called nodes. Such a graph is called a circular planar electrical network. A grove is a spanning forest in which every component contains at least one node. The connected components of a grove determine a partition of the nodes. We relate weighted grove counts to invariant theory for the symplectic group. To a planar electrical network $G$, we associate an $\mathrm{Sp}(2n)$-invariant tensor $Z_G$. For $\mathrm{Sp}(2)=\mathrm{SL}(2)$, we expand $Z_G$ in the Temperley--Lieb basis indexed by noncrossing matchings and relate its coefficients to the Kenyon--Wilson grove formulas. For $\mathrm{Sp}(4)$, we give reduction rules for superpositions of two groves and prove that the tensors indexed by $3$-noncrossing matchings form a basis of the space of $\mathrm{Sp}(4)$-invariant tensors. The coefficients of $Z_G$ in this basis are weighted counts of reduced double groves, up to normalization.
