Artin群中的回文与序
Palindromes and orderings in Artin groups
摘要：
辫群$B_{n}$，赋予阿廷的介绍，承认两个杰出的对合。一种是反自同构${\rm{rev}}: B_{n} \to B_{n}$，即$v \mapsto \bar{v}$，它是通过以相反的顺序（从右到左而不是从左到右）阅读辫子来定义的。另一种是广义半扭转共轭$τ:x \mapsto Δ^{-1}x Δ$（Garside元）。更一般地说，对合${\rm{rev}}$是为所有的阿丁群（具有阿丁表示）定义的，而对合$τ$是为所有的有限型阿丁群定义的。回文是在rev下不变的元素。我们对有限型Artin群中的回文和在$τ$下不变的回文进行了分类。工具是初等重写和建设明确的左序兼容修订。最后，我们讨论了推广到Artin群的无限型和Garside群。
英文摘要：
The braid group $B_{n}$, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism ${\rm{rev}}: B_{n} \to B_{n}$, $v \mapsto \bar{v}$, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation $τ:x \mapsto Δ^{-1}x Δ$ by the generalized half-twist (Garside element). More generally, the involution ${\rm{rev}}$ is defined for all Artin groups (equipped with Artin's presentation) and the involution $τ$ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We classify palindromes and palindromes invariant under $τ$ in Artin groups of finite type. The tools are elementary rewriting and the construction of explicit left-orderings compatible with rev. Finally, we discuss generalizations to Artin groups of infinite type and Garside groups.
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