A note on involution prefixes in Coxeter groups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866929695007703040 |
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| author | Hart, Sarah B. Rowley, Peter J. |
| author_facet | Hart, Sarah B. Rowley, Peter J. |
| contents | Let $(W, R)$ be a Coxeter system and let $w \in W$. We say that $u$ is a prefix of $w$ if there is a reduced expression for $u$ that can be extended to one for $w$. That is, $w = uv$ for some $v$ in $W$ such that $\ell(w) = \ell(u) + \ell(v)$. We say that $w$ has the ancestor property if the set of prefixes of $w$ contains a unique involution of maximal length. In this paper we show that all Coxeter elements of finitely generated Coxeter groups have the ancestor property, and hence a canonical expression as a product of involutions. We conjecture that the property in fact holds for all non-identity elements of finite Coxeter groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00777 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on involution prefixes in Coxeter groups Hart, Sarah B. Rowley, Peter J. Group Theory 20F55 Let $(W, R)$ be a Coxeter system and let $w \in W$. We say that $u$ is a prefix of $w$ if there is a reduced expression for $u$ that can be extended to one for $w$. That is, $w = uv$ for some $v$ in $W$ such that $\ell(w) = \ell(u) + \ell(v)$. We say that $w$ has the ancestor property if the set of prefixes of $w$ contains a unique involution of maximal length. In this paper we show that all Coxeter elements of finitely generated Coxeter groups have the ancestor property, and hence a canonical expression as a product of involutions. We conjecture that the property in fact holds for all non-identity elements of finite Coxeter groups. |
| title | A note on involution prefixes in Coxeter groups |
| topic | Group Theory 20F55 |
| url | https://arxiv.org/abs/2502.00777 |