链接系统的分类
Classification of linkage systems
摘要：
通过添加额外的根$γ$，从卡特图$Γ$获得链接图，使得所得的根的子集是线性无关的。对于每个连锁图，我们将连锁标签矢量$γ^{\nabla}$与Dynkin标签相关联。在$W^{\vee}_{S}$组的作用下连接的连接图构成连接系统$\mathscr{L}(Γ)$。对于任意的单系卡特图，构造系统$\mathscr{L}(Γ)$。为了获得连锁图$θ^{\nabla}$，我们使用一个容易验证的标准：$\mathscr{B}^{\vee}_Γ(θ^{\nabla}) < 2$，其中$\mathscr{B}^{\vee}_Γ$是与$Γ$相关的逆二次型。Dynkin图$Γ'$使得秩（$Γ'$）=秩（$Γ$）+1并且任何与$Γ$相关联的根子集$S$位于$\varPhi(Γ')$中，被称为Dynkin扩展。连接系统$\mathscr{L}(Γ)$是$Γ_i$-$Γ<_D Γ_i$的所有Dynkin扩展所采用的组件$\mathscr{L}_{Γ_i}(Γ)$的联合。$\varPhi(Γ')$的根的子集$\varPhi(S)$，线性依赖于$S$的根，被称为部分根系。$\mathscr{L}_{Γ'}(Γ)$的大小估计如下：$|\mathscr{L}_{Γ'}(Γ)| \leq |\varPhi(Γ')| - |\varPhi(S)|$。卡特图解$E_l$和$E_l(a_i)$（分别为$D_l$和$D_l(a_k)$）被认为是共价的。对于任意一对共价Carter图{ $Γ, \widetildeΓ$ }，其中$Γ$是Dynkin图，我们显式地构造可逆线性映射$M : \mathcal{P} \longrightarrow \mathcal{R}$，其中$\mathcal{R}$（resp. $\mathcal{P}$）是根系（resp.部分根系）对应于$Γ$（分别为$\widetildeΓ$）。特别是，我们有$|\mathscr{L}(\widetildeΓ)| = |\mathscr{L}(Γ)|$。
英文摘要：
A linkage diagram is obtained from the Carter diagram $Γ$ by adding an extra root $γ$, so that the resulting subset of roots is linearly independent. With every linkage diagram we associate the linkage label vector $γ^{\nabla}$, similar to Dynkin labels. The linkage diagrams connected under the action of the group $W^{\vee}_{S}$ constitute the the linkage system $\mathscr{L}(Γ)$. For any simply-laced Carter diagram, the system $\mathscr{L}(Γ)$ is constructed. To obtain linkage diagrams $θ^{\nabla}$, we use an easily verifiable criterion: $\mathscr{B}^{\vee}_Γ(θ^{\nabla}) < 2$, where $\mathscr{B}^{\vee}_Γ$ is the inverse quadratic form associated with $Γ$. A Dynkin diagram $Γ'$ such that rank($Γ'$) = rank($Γ$) + 1 and any $Γ$-associated root subset $S$ lies in $\varPhi(Γ')$, is said to be the Dynkin extension. The linkage system $\mathscr{L}(Γ)$ is the union of $Γ_i$-components $\mathscr{L}_{Γ_i}(Γ)$ taken for all Dynkin extensions of $Γ<_D Γ_i$. The subset $\varPhi(S)$ of roots of $\varPhi(Γ')$, linearly dependent on roots of $S$ is said to be a partial root system. The size of $\mathscr{L}_{Γ'}(Γ)$ is estimated as follows: $|\mathscr{L}_{Γ'}(Γ)| \leq |\varPhi(Γ')| - |\varPhi(S)|$. Carter diagrams $E_l$ and $E_l(a_i)$ (resp. $D_l$ and $D_l(a_k)$) are said to be covalent. For any pair {$Γ, \widetildeΓ$} of covalent Carter diagrams, where $Γ$ is the Dynkin diagram, we explicitly construct the invertible linear map $M : \mathcal{P} \longrightarrow \mathcal{R}$, where $\mathcal{R}$ (resp. $\mathcal{P}$) is the root system (resp. partial root system) corresponding to $Γ$ (resp. $\widetildeΓ$). In particular, we have $|\mathscr{L}(\widetildeΓ)| = |\mathscr{L}(Γ)|$.
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