网络世界、网络着色矩阵和网络混合矩阵
Web worlds, web-colouring matrices, and web-mixing matrices
摘要：
我们引入了一个新的组合对象称为网络世界，由一组网络图。网络世界的图是图的概括，每个图都建立在相同的底层图上。图中没有普通的顶点，而是有桩，与桩相连的边在桩上有不同的高度。Web图表的Web世界是所有Web图表的集合，这些图表是通过排列边的端点在peg上出现的顺序而产生的。动机来自粒子物理学，其中网络图作为描述非阿贝尔规范（杨-米尔斯）理论中散射振幅的特殊类型的费曼图出现。对于每个网络世界，我们关联两个矩阵，称为网络着色矩阵和网络混合矩阵。这些矩阵的元素由网状图（D_1，D_2）的序偶对索引，并由D_1的边的着色计算，这些着色在由每个着色确定的变换下产生D_2。我们表明，着色的网络图（其组成不可分解的图都是唯一的），导致重建的图是等价于保序映射的某些偏序集（偏序集），可以从网络图。对于其网络图的所有边标签都等于1的网络世界，网络混合矩阵和网络着色矩阵的对角项通过对由相关偏序集的所有线性扩展的Jordan-Holder集的排列中的下降确定的某些多项式求和来获得。我们根据三个统计量推导出网络世界数量的三变量生成函数，并列举出一个网络世界中不同网络图的数量。三个特殊的网络世界进行了详细的研究，并在每种情况下计算的网络混合矩阵的痕迹。
英文摘要：
We introduce a new combinatorial object called a web world that consists of a set of web diagrams. The diagrams of a web world are generalizations of graphs, and each is built on the same underlying graph. Instead of ordinary vertices the diagrams have pegs, and edges incident to a peg have different heights on the peg. The web world of a web diagram is the set of all web diagrams that result from permuting the order in which endpoints of edges appear on a peg. The motivation comes from particle physics, where web diagrams arise as particular types of Feynman diagrams describing scattering amplitudes in non-Abelian gauge (Yang-Mills) theories. To each web world we associate two matrices called the web-colouring matrix and web-mixing matrix. The entries of these matrices are indexed by ordered pairs of web diagrams (D_1,D_2), and are computed from those colourings of the edges of D_1 that yield D_2 under a transformation determined by each colouring. We show that colourings of a web diagram (whose constituent indecomposable diagrams are all unique) that lead to a reconstruction of the diagram are equivalent to order-preserving mappings of certain partially ordered sets (posets) that may be constructed from the web diagrams. For web worlds whose web graphs have all edge labels equal to 1, the diagonal entries of web-mixing and web-colouring matrices are obtained by summing certain polynomials determined by the descents in permutations in the Jordan-Holder set of all linear extensions of the associated poset. We derive tri-variate generating generating functions for the number of web worlds according to three statistics and enumerate the number of different web diagrams in a web world. Three special web worlds are examined in great detail, and the traces of the web-mixing matrices calculated in each case.
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