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

  • Authors:

  • Mark Dukes;Einan Gardi;Einar SteingríMsson;Chris D. White

  • Affiliations:

  • Department of Computer and Information Sciences, University of Strathclyde, Glasgow, G1 1XH, UK;School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3JZ, UK;Department of Computer and Information Sciences, University of Strathclyde, Glasgow, G1 1XH, UK;School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2013

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Abstract

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 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.