根系和图解演算。III。连杆图的半Coxeter轨道与Carter定理
Root systems and diagram calculus. III. Semi-Coxeter orbits of linkage diagrams and the Carter theorem
摘要：
从卡特图$Γ$通过将一个根与它的键加在一起使得所得到的根的子集是线性无关的而得到的图被称为{\it连接图}。给定一个链接图，我们关联链接标签向量，该向量类似于Dynkin标签向量。类似于对偶Weyl群，我们引入与$Γ$相关联的群$W^{\vee}_L$，我们称它为对偶部分Weyl群。在$W^{\vee}_L$作用下连接的连接标签向量构成连接系统$\mathscr{L}(Γ)$，它类似于半单李代数的表示理论中出现的权系统。卡特定理指出外尔群$W$的每个元素都可以表示为两个对合的乘积。本文在描述连锁系统$\mathscr{L}(Γ)$和任意Carter图$Γ$的连锁标号向量的半Coxeter轨道的基础上，给出了这一定理的证明.证明的主要思想是基于这样一个事实，即除了少数例外，在每个半Coxeter轨道中都有一个特殊的链接图-称为{\it单色}，其中分解为两个对合的乘积是平凡的。
英文摘要：
A diagram obtained from the Carter diagram $Γ$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent is said to be the {\it linkage diagram}. Given a linkage diagram, we associate the linkage labels vector, which is introduced like the vector of Dynkin labels. Similarly to the dual Weyl group, we introduce the group $W^{\vee}_L$ associated with $Γ$, and we call it the dual partial Weyl group. The linkage labels vectors connected under the action of $W^{\vee}_L$ constitute the linkage system $\mathscr{L}(Γ)$, which is similar to the weight system arising in the representation theory of the semisimple Lie algebras. The Carter theorem states that every element of a Weyl group $W$ is expressible as the product of two involutions. We give the proof of this theorem based on the description of the linkage system $\mathscr{L}(Γ)$ and semi-Coxeter orbits of linkage labels vectors for any Carter diagram $Γ$. The main idea of the proof is based on the fact that, with a few exceptions, in each semi-Coxeter orbit there is a special linkage diagram -- called {\it unicolored}, for which the decomposition into the product of two involutions is trivial.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
