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

Mark Dukes, Einan Gardi, Einar Steingrimsson, Chris D. White

TL;DR

This work introduces web worlds and web diagrams as a combinatorial framework inspired by non-Abelian gauge theory, defining two key matrices, the web-colouring matrix ${\mathfrak{M}}^{(W)}(x)$ and the web-mixing matrix ${\mathfrak{R}}^{(W)}$, to study colourings and reconstructions of diagrams. It establishes a deep connection between colourings that uniquely reconstruct a diagram and order-preserving maps of the associated decomposition poset, allowing diagonal entries of the matrices to be expressed via descent statistics on linear extensions, and it develops tri-variate generating functions to count web worlds by three statistics. The paper provides explicit calculations for three detailed web-world cases, revealing how traces of ${\mathfrak{R}}^{(W)}$ are governed by permutation statistics, and it highlights a profound link between transitive web worlds and $(2+2)$-free posets. By connecting combinatorial structures (posets, Eulerian numbers, Stirling numbers) with physics-mourced web diagrams, the authors open multiple directions for exact solvability, enumeration, and potential applications to scattering amplitudes in quantum chromodynamics.

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 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.

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

TL;DR

This work introduces web worlds and web diagrams as a combinatorial framework inspired by non-Abelian gauge theory, defining two key matrices, the web-colouring matrix and the web-mixing matrix , to study colourings and reconstructions of diagrams. It establishes a deep connection between colourings that uniquely reconstruct a diagram and order-preserving maps of the associated decomposition poset, allowing diagonal entries of the matrices to be expressed via descent statistics on linear extensions, and it develops tri-variate generating functions to count web worlds by three statistics. The paper provides explicit calculations for three detailed web-world cases, revealing how traces of are governed by permutation statistics, and it highlights a profound link between transitive web worlds and -free posets. By connecting combinatorial structures (posets, Eulerian numbers, Stirling numbers) with physics-mourced web diagrams, the authors open multiple directions for exact solvability, enumeration, and potential applications to scattering amplitudes in quantum chromodynamics.

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 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.

Paper Structure

This paper contains 10 sections, 21 theorems, 79 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Web-diagrams
  3. Self-reconstructing colourings and order-preserving maps
  4. The number of web worlds
  5. A brief interlude on colouring sequences
  6. Case 1: A web world with $\mathrm{Pegs}(D)=(1,1,\ldots,1,n)$
  7. Case 2: A web world with $\mathrm{Pegs}(D)=(1,2,\ldots,2,1)$
  8. Case 3: A web world with $\mathrm{Pegs}(D)=(2,2,\ldots,2)$
  9. Transitive web worlds
  10. Further comments and open problems

Key Result

Theorem 2.13

GPO Let $W$ be a web world.

Figures (7)

  • Figure 1: In the diagram on the left the indices of the pegs are shown at the bottom. The heights of the endpoints of the edges are shown in italics at each endpoint. The unique edge between pegs 3 and 6 is represented by the 4-tuple $(3,6,2,4)$ since the left endpoint of the edge (on peg 3) has height 2 and the right endpoint of the edge (on peg 6) has height 4. The diagram on the right is the Feynman diagram illustration of the web diagram.
  • Figure 2: The transformation $X\to \mathop{\mathrm{\mathsf{rel}}}\nolimits(X)$.
  • Figure 3: The four web diagrams in the web world generated from $D$ in Example \ref{['falkirk']}.
  • Figure 4: Example of a web diagram in a $\mathrm{Pegs}(D) = (1,1,1,1,1,1,6)$ web world. This is represented by $D_{\pi}$ in Equation (\ref{['deepi']}) with $\pi=(5,4,2,1,6,3)$.
  • Figure 5: An example of a web diagram of the form given in Equation (\ref{['casetwotype']}).
  • ...and 2 more figures

Theorems & Definitions (60)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 50 more