论文标题
Weyl Chambers短期四分之一平面晶格路径
Weyl chambers for short step Quarter-plane Lattice Paths
论文作者
论文摘要
我们考虑了局限于四分之一平面的短步格路径的四个例子。这些是Kreweras，反向Kreweras，Gessel和Mishna-Rechnitzer晶格路径。反向Kreweras可以简单地解决，因此有趣的是与Kreweras路径和Gessel路径形成鲜明对比，因为后者两个在历史上在历史上很难解决。 Mishna-Rechnitzer路径很有趣，因为它们与无限级别组相关。我们将通过考虑与它们的步骤集相关的Weyl腔室来对所有这些特性进行一些几何见解。对于反向kreweras路径，Weyl腔室壁与四分之一平面边界一致，因此Bethe Ansatz或使用Gessel-Zeilberger theorem可以很容易地解决问题。对于Kreweras路径，四分之一平面对应于两个相邻的Weyl腔室的结合，因此，Bethe Ansatz和Gessel-Zeilberger定理都不适用，这使得问题更加难以解决。同样，Gessel路径的四分之一平面是三个Weyl腔室的结合。对于Mishna-Rechnitzer路径，步骤集具有非零的Barycenter，导致二面反射组。仿射结构对应于随机步行中的漂移。四分之一平面是无限数量的Weyl Alcoves的结合。
We consider four examples of short step lattice paths confined to the quarter plane. These are the Kreweras, Reverse Kreweras, Gessel, and Mishna-Rechnitzer lattice paths.The Reverse Kreweras are straightforward to solve and thus interesting as a contrast to the Kreweras paths and Gessel paths as the latter two have historically been significantly more difficult to solve. The Mishna-Rechnitzer paths are interesting as they are associated with an infinite order group. We will give some geometrical insight into all these properties by considering the Weyl chambers associated with their step sets.For Reverse Kreweras paths the Weyl chamber walls coincide with the quarter plane boundary and hence the problem is readily solvable by Bethe Ansatz or by using the Gessel-Zeilberger Theorem. For Kreweras paths the quarter plane corresponds to the union of two adjacent Weyl Chambers and hence neither the Bethe Ansatz nor the Gessel-Zeilberger Theorem are directly applicable making the problem considerably more difficult to solve. Similarly, the quarter plane for Gessel paths is the union of three Weyl chambers. For Mishna-Rechnitzer paths the step set has non-zero barycenter leading to an affine dihedral reflection group. The affine structure corresponds to the drift in the random walk. The quarter plane is the union of an infinite number of Weyl alcoves.
