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Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes

Fadi Awik, Jadyn Breland, Quentin Cadman, Dana C. Ernst

Abstract

In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, we use the decomposition to prove that braid graphs are partial cubes, i.e., can be isometrically embedded into a hypercube. For a special class of links, called Fibonacci links, we prove that the corresponding braid graphs are Fibonacci cubes.

Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes

Abstract

In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, we use the decomposition to prove that braid graphs are partial cubes, i.e., can be isometrically embedded into a hypercube. For a special class of links, called Fibonacci links, we prove that the corresponding braid graphs are Fibonacci cubes.

Paper Structure

This paper contains 7 sections, 28 theorems, 47 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Architecture of braid classes in simply-laced Coxeter systems
  4. Classification of links and braid graphs in Coxeter systems of type $A_n$
  5. Braid graphs as partial cubes
  6. Fibonacci links and Fibonacci cubes
  7. Closing

Key Result

Proposition 2.3

In a Coxeter system $(W,S)$, any two reduced expressions for the same group element differ by a sequence of commutation and braid moves.

Figures (11)

  • Figure 1: Examples of common simply-laced Coxeter graphs.
  • Figure 2: Matsumoto graph for the longest element in $W(A_3)$.
  • Figure 3: Braid graphs generated by various reduced expressions.
  • Figure 4: Braid graph for the reduced expression from Example \ref{['ex:linkfactorization']} and its decomposition into a box product of braid graphs for the corresponding link factors.
  • Figure 5: Braid graph for the reduced expression from Example \ref{['ex:productoflollipops']} and its decomposition into a box product of braid graphs for the corresponding link factors.
  • ...and 6 more figures

Theorems & Definitions (79)

  • Example 2.1
  • Example 2.2
  • Proposition 2.3: Matsumoto's Theorem
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Definition 3.4
  • ...and 69 more