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Web worlds, web-colouring matrices, and web-mixing matrices

Dukes, Mark and Gardi, Einan and Steingrimsson, Einar and White, Chris (2013) Web worlds, web-colouring matrices, and web-mixing matrices. Journal of Combinatorial Theory Series A, 120 (5). pp. 1012-1037. ISSN 0097-3165

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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