Introduction to Representation Theory
《Introduction to Representation Theory》 · 印刷第 160—161 页 Section 6.4 的图、内积和单根约定；第 163 页 Definition 6.4.10 的简单反射定义；第 169 页 Section 6.7「Coxeter elements」，Definition 6.7.1 完整定义
From now on, let $\Gamma$ be a fixed graph of type $A_n$, $D_n$, $E_6$, $E_7$, $E_8$. We denote the adjacency matrix of $\Gamma$ by $R_\Gamma$. Definition 6.4.1 (Cartan matrix). We define the Cartan matrix of $\Gamma$ as $A_\Gamma=2\operatorname{Id}-R_\Gamma$. On the lattice $\mathbb Z^n$ (or the space $\mathbb R^n$) we then define an inner product $B(x,y)=x^TA_\Gamma y$ corresponding to the graph $\Gamma$. […] Definition 6.4.5. We call vectors of the form $\alpha_i=(0,\ldots,\overbrace{1}^{i\text{th}},\ldots,0)$ simple roots. […] Definition 6.4.10. Let $\alpha\in\mathbb Z^n$ be a positive root. The reflection $s_\alpha$ is defined by the formula $s_\alpha(v)=v-B(v,\alpha)\alpha$. We denote $s_{\alpha_i}$ by $s_i$ and call these simple reflections. […] Definition 6.7.1. Let $Q$ be a quiver and let $\Gamma$ be the underlying graph. Fix any labeling $1,\ldots,n$ of the vertices of $\Gamma$. Then the Coxeter element $c$ of $Q$ corresponding to this labeling is defined as $c=s_1s_2\cdots s_n$.