论文标题
旋转旋转诱导平面图上唯一一致的沙珀托托尔结构
Rotor-Routing Induces the Only Consistent Sandpile Torsor Structure on Plane Graphs
论文作者
论文摘要
我们做出准确的态度，并证明克里夫人对沙珀群体在跨越树木上的作用进行了猜想。更具体地说，猜想指出，在平面图上存在独特的``适当'''适当的''sandpile torsor结构，该结构是由旋转路线引起的。 首先，我们严格地将沙珀torsor算法（在平面图上）定义为将每个平面图（即平面图与适当的色带结构）与其在其跨越树上的砂型组的自由性传递作用相关联的地图。然后，我们定义了一个一致性的概念，该概念要求将Torsor算法保留在某种类别的收缩和删除方面。使用这些定义，我们表明转子穿孔的沙台托索尔算法是一致的。此外，我们证明了平面图上只有其他三种一致的算法，它们的结构与转子线相同。 我们还在常规矩阵上定义了沙珀torsor算法，并在这种情况下提出了一致性的概念。我们猜想，备用者YUEN算法是一致的，并且在常规矩形上只有其他三种一致的沙翼torsor算法，所有这些算法都具有相同的结构。
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.
