# What is the relationship between Cartan matrices and Coxeter matrices?

Let $\Phi$ be a crystallographic root system. The Cartan matrix of $\Phi$ is the integer matrix $A=(a_{ij})$, where $a_{ij}=\frac{2(\alpha_i,\ \beta_j)}{(\alpha_i,\ \alpha_j)}$ for all $\alpha_i,\alpha_j \in \Phi$, and $i,j \in I$.

A group $W$ is said to be a Coxeter group if it has a presentation of the form \begin{align} W= \langle s_i:i\in I |(s_i s_j)^{m(i,j)}=1,for\ all\ i,j \in I\rangle. \end{align} If $W=W_{\Phi}$ is a crystallographic reflection group with simple system $\alpha_1,\alpha_2,\ldots \alpha_n$ ,we note that for $i \neq j$, if $m(i,j)=2$, then $a_{ij}=a_{ji}=0$; if $m(i,j)=3$, then $a_{ij}=a_{ji}=-1$; if $m(i,j)=4$, then $a_{ij}=-1,a_{ji}=-2$; if $m(i,j)=6$, then $a_{ij}=-1,a_{ji}=-3$.

Is this the relationship between them?

Take $m_{ij}$ such that $(\alpha_i, \alpha_j) = - |\alpha_i||\alpha_j|\cos(\pi/m_{ij})$, where $|\alpha_i|=\sqrt{(\alpha_i,\alpha_i)}$. Then $(m_{ij})$ is the Coxeter matrix. Take $c_{ij}=2(\alpha_i,\alpha_j)/(\alpha_j,\alpha_j)$. Then $(c_{ij})$ is the Cartan matrix.

For example, when $(c_{ij})=\left( \begin{matrix} 2 & -1 \\ -1 & 2 \end{matrix} \right)$. We have $(\alpha_1, \alpha_2) = -1/2$, $|\alpha_1|=|\alpha_2|=1$. Therefore $m_{12}=3$. Similary, $m_{21}=3, m_{11}=m_{22}=1$. We recover the Coxeter matrix of type $A_2$: $\left( \begin{matrix} 1 & 3 \\ 3 & 1 \end{matrix} \right)$.

When $(c_{ij})=\left( \begin{matrix} 2 & -2 \\ -1 & 2 \end{matrix} \right)$. We have $2(\alpha_1, \alpha_2)/(\alpha_2,\alpha_2) = -2$, $|\alpha_1|=\sqrt{2},|\alpha_2|=1$. Therefore $(\alpha_1,\alpha_2)=-1$. Using the formula $(\alpha_i, \alpha_j) = - |\alpha_i||\alpha_j|\cos(\pi/m_{ij})$, we have $-1=(\alpha_1,\alpha_2)=- |\alpha_1||\alpha_2|\cos(\pi/m_{12})$, $m_{12}=4$.

We have $-1=(\alpha_2,\alpha_1)=- |\alpha_2||\alpha_1|\cos(\pi/m_{21})$, $m_{21}=4$.

Similary, $m_{11}=m_{22}=1$. We recover the Coxeter matrix of type $B_2$: $\left( \begin{matrix} 1 & 4 \\ 4 & 1 \end{matrix} \right)$.

Dynkin diagram dertermines the Cartan matrix of crystallographic root system $\Phi$, that is, Dynkin diagram is a graph with vertices $I$ with $a_{ij}a_{ji}$ edges between $i$ and $j$.

Moreover, Dynkin diagram implies the coefficient $m(i,j)$ of $s_is_j$, see http://arxiv.org/pdf/1112.2300.pdf.