Barnes, Jillian
(2022)
*Structural properties of braid graphs in simply-laced triangle-free Coxeter systems.*
Masters thesis, Northern Arizona University.

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## Abstract

Any two reduced expressions for the same Coxeter group element are related by a sequence of commutation and braid moves. We say that two reduced expressions are braid equivalent if they are related via a sequence of braid moves, and the corresponding equivalence classes are called braid classes. Each braid class can be encoded in terms of a braid graph in a natural way. In a recent paper, Awik et al. proved that every reduced expression in a simply-laced Coxeter group 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. Moreover, the authors proved that when the Coxeter system is triangle free (i.e., the corresponding Coxeter graph has no three-cycles), the braid graph for a reduced expression is a partial cube (i.e., isometric to a sub- graph of a hypercube). In this thesis, we study the structural properties of braid classes in simply-laced triangle-free Coxeter systems. In particular, we provide precise information about the local structure of reduced expressions in the braid class for a link and produce an alternate proof of the fact that every braid graph in simply-laced triangle-free Coxeter systems is a partial cube. Moreover, we outline the obstructions to proving the conjectures that every braid graph in a simply-laced triangle-free Coxeter system is median and corresponds to the Hasse diagram for a distributive lattice.

Item Type: | Thesis (Masters) |
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Publisher’s Statement: | © Copyright is held by the author. Digital access to this material is made possible by the Cline Library, Northern Arizona University. Further transmission, reproduction or presentation of protected items is prohibited except with permission of the author. |

Keywords: | Coxeter group; Braid classes; Braid graphs; Hasse diagram |

Subjects: | Q Science > QA Mathematics |

NAU Depositing Author Academic Status: | Student |

Department/Unit: | Graduate College > Theses and Dissertations College of the Environment, Forestry, and Natural Sciences > Mathematics and Statistics |

Date Deposited: | 14 Jul 2022 17:03 |

Last Modified: | 14 Jul 2022 17:03 |

URI: | https://openknowledge.nau.edu/id/eprint/5845 |

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