Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes
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arXiv
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| Format: | Preprint |
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2021
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| author | Awik, Fadi Breland, Jadyn Cadman, Quentin Ernst, Dana C. |
| author_facet | Awik, Fadi Breland, Jadyn Cadman, Quentin Ernst, Dana C. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_12318 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes Awik, Fadi Breland, Jadyn Cadman, Quentin Ernst, Dana C. Combinatorics Group Theory 20F55, 05C60, 05E15, 05A05 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. |
| title | Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes |
| topic | Combinatorics Group Theory 20F55, 05C60, 05E15, 05A05 |
| url | https://arxiv.org/abs/2104.12318 |