Coxeter systems, left inversion sets, and higher dimensional cubes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912310843408384 |
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| author | Gimenez, Harrison |
| author_facet | Gimenez, Harrison |
| contents | Let $ (W,S)$ be a Coxeter system. We investigate the equation $ w(Φ_{x}) = Φ_{y}$ where $ w,x,y\in W$ and $ Φ_{x}$, $Φ_{y}$ denote the left inversion sets of $ x$ and $ y$. We then define a commutative square diagram called a Coxeter square which describes the relationship between 4 non-identity elements of the Coxeter group $ W$ and the equation $ w(Φ_{x}) = Φ_{y}$. Coxeter squares were first introduced by Dyer, Wang in \cite{dyer2011groupoids2} and \cite{dyer2019characterization}. Coxeter squares can be \textquotedblleft glued" together by compatible edges to form commutative diagrams in the shape of higher dimensional cubes called Coxeter $n$-cubes, which were first defined by Dyer in Example 12.5 of \cite{dyer2011groupoids2}. When $ |W| < \infty$ and $ |S| = n$, we show that Coxeter $n$-cubes must exist within $ (W,S)$. We then prove results about Coxeter $n$-cubes in the $A_{n}$ Coxeter system. We establish an explicit bijection between Coxeter $n$-cubes (modulo orientation) in $ A_{n}$ and binary trees with $n+1$ leaves. We also show that an element $x$ of $ A_{n}$ appears as the edge of some Coxeter $n$-cube if and only if $ x$ is a bigrassmannian permutation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03911 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coxeter systems, left inversion sets, and higher dimensional cubes Gimenez, Harrison Group Theory Combinatorics 20F55 (Primary) 17B22, 20L05 (Secondary) Let $ (W,S)$ be a Coxeter system. We investigate the equation $ w(Φ_{x}) = Φ_{y}$ where $ w,x,y\in W$ and $ Φ_{x}$, $Φ_{y}$ denote the left inversion sets of $ x$ and $ y$. We then define a commutative square diagram called a Coxeter square which describes the relationship between 4 non-identity elements of the Coxeter group $ W$ and the equation $ w(Φ_{x}) = Φ_{y}$. Coxeter squares were first introduced by Dyer, Wang in \cite{dyer2011groupoids2} and \cite{dyer2019characterization}. Coxeter squares can be \textquotedblleft glued" together by compatible edges to form commutative diagrams in the shape of higher dimensional cubes called Coxeter $n$-cubes, which were first defined by Dyer in Example 12.5 of \cite{dyer2011groupoids2}. When $ |W| < \infty$ and $ |S| = n$, we show that Coxeter $n$-cubes must exist within $ (W,S)$. We then prove results about Coxeter $n$-cubes in the $A_{n}$ Coxeter system. We establish an explicit bijection between Coxeter $n$-cubes (modulo orientation) in $ A_{n}$ and binary trees with $n+1$ leaves. We also show that an element $x$ of $ A_{n}$ appears as the edge of some Coxeter $n$-cube if and only if $ x$ is a bigrassmannian permutation. |
| title | Coxeter systems, left inversion sets, and higher dimensional cubes |
| topic | Group Theory Combinatorics 20F55 (Primary) 17B22, 20L05 (Secondary) |
| url | https://arxiv.org/abs/2504.03911 |