论文标题
间隔组与有限的Coxeter组，第二部分
Interval groups related to finite Coxeter groups, Part II
论文作者
论文摘要
我们提供了与有限的Coxeter组中与准氧气元素有关的间隔组的演示的完整描述。在简单的情况下，我们表明，每个间隔组都是通过一组扭曲的循环换向器的正常闭合与相应的Carter图相关的ARTIN组的商，每个图的每个4周期都为一个。我们的技术还可以为有限的Coxeter组的Artin组提供了类似的结果，该组是与Coxeter元素相对应的间隔组。我们还分析了发现新的Garside结构的非简单系列案例中的情况。 此外，我们获得了我们认为的间隔组是否是同构的完整分类。实际上，使用山雀方法，我们证明了适当的准氧气元素的间隔组对同一类型的Artin组而言并非同构，而在$ n $的情况下，或在任何特殊情况下，$ n $均为$ d_n $。在[BHNR22]中，我们使用不同的方法显示了此结果适用于所有$ n \ geq 4 $的类型$ d_n $。
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 $D_n$ when $n$ is even or in any of the exceptional cases. In [BHNR22], we show using different methods that this result holds for type $D_n$ for all $n \geq 4$.
