Rotor-Routing 在平面图上诱导唯一的 Consistent Sandpile Torsor 结构
Rotor-Routing Induces the Only Consistent Sandpile Torsor Structure on Plane Graphs
AI总结：
本文证明 Klivans 猜想，严格定义并证明了平面图上由 rotor-routing 诱导的唯一一致的沙堆挠子结构，并将其扩展至正则拟阵，提出 Backman-Baker-Yuen 算法的一致性猜想。
摘要：
我们明确表述并证明了 Klivans 关于沙堆群在生成树上作用的一个猜想。更具体地，该猜想指出在平面图上存在唯一的“足够好”的沙堆挠子结构，该结构由 rotor-routing 诱导。首先，我们严格定义了平面图上的沙堆挠子算法，即将每个平面图（即具有适当带状结构的平面图）与其沙堆群在其生成树上的自由传递作用相关联的映射。然后，我们定义了一致性的概念，要求挠子算法在某一类收缩和删除操作下保持不变。利用这些定义，我们证明了 rotor-routing 沙堆挠子算法是一致的。此外，我们证明了在平面图上仅存在另外三种一致的算法，它们与 rotor-routing 具有相同的结构。我们还定义了正则拟阵上的沙堆挠子算法，并在此背景下提出了一致性的概念。我们猜想 Backman-Baker-Yuen 算法是一致的，且在正则拟阵上仅存在另外三种一致的沙堆挠子算法，它们均具有相同的结构。
英文摘要：
We make precise and prove a conjecture of Klivans about actions of the sandpile group on spanning trees. More specifically, the conjecture states that there exists a unique ``suitably nice'' sandpile torsor structure on plane graphs which is induced by rotor-routing. First, we rigorously define a sandpile torsor algorithm (on plane graphs) to be a map which associates each plane graph (i.e., planar graph with an appropriate ribbon structure) with a free transitive action of its sandpile group on its spanning trees. Then, we define a notion of consistency, which requires a torsor algorithm to be preserved with respect to a certain class of contractions and deletions. Using these definitions, we show that the rotor-routing sandpile torsor algorithm is consistent. Furthermore, we demonstrate that there are only three other consistent algorithms on plane graphs, which all have the same structure as rotor-routing. We also define sandpile torsor algorithms on regular matroids and suggest a notion of consistency in this context. We conjecture that the Backman-Baker-Yuen algorithm is consistent, and that there are only three other consistent sandpile torsor algorithms on regular matroids, all with the same structure.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
