Barbara Baumeister, Derek F. Holt, Georges Neaime, Sarah Rees
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引用次数: 0
摘要
摘要给出了有限Coxeter群中与拟Coxeter元相关的区间群的完整描述。在简单加权的情况下,我们证明了每个区间群都是Artin群的商,通过一组扭环换向子的正规闭包与相应的卡特图相关联,图的每4个环对应一个扭环换向子。我们的方法也证明了有限Coxeter群的Artin群的一个类似结果,这些群是对应于Coxeter元的区间群。我们还分析了在非简单的情况下,一个新的Garside结构被发现的情况。进一步,我们得到了所考虑的区间群是否与相关的Artin群同构的完全分类。事实上,我们利用Tits的方法证明了在当为偶数或任何例外情况下,适当拟- Coxeter元的区间群与同类型的Artin群不同构。在Baumeister et al. (J. Algebra 629(2023), 399-423)中,我们使用不同的方法表明该结果适用于所有类型。
Interval groups related to finite Coxeter groups Part II
Abstract We provide a complete description of the presentations of the interval groups related to quasi‐Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associated with the corresponding Carter diagram by the normal closure of a set of twisted cycle commutators, one for each 4‐cycle of the diagram. Our techniques also reprove an analogous result for the Artin groups of finite Coxeter groups, which are interval groups corresponding to Coxeter elements. We also analyse the situation in the non‐simply laced cases, where a new Garside structure is discovered. Furthermore, we obtain a complete classification of whether the interval group we consider is isomorphic or not to the related Artin group. Indeed, using methods of Tits, we prove that the interval groups of proper quasi‐Coxeter elements are not isomorphic to the Artin groups of the same type, in the case of when is even or in any of the exceptional cases. In Baumeister et al. (J. Algebra 629 (2023), 399–423), we show using different methods that this result holds for type for all .