the simple roots $\alpha_i$: $$\begin{gathered}
\l\alpha_i,\w_j\r=\delta_{ij},\quad 1\leq i,j\leq n.\end{gathered}$$ It is also a basis in the subspace ${\mathcal H}\subset\R^{n+1}$ (see (\[plane H\])). The bases $\alpha$ and $\w$ are related by the Cartan matrix: $$\begin{gathered}
\alpha=\mathfrak{C}\w,\quad \w=\mathfrak{C}^{-1}\alpha.\end{gathered}$$

Throughout the paper, we use $\lambda\in\mathcal H$. Here, we fix the notation for its coordinates relative to the $e$- and $\omega$-bases: $$\begin{gathered}
\label{notation_bases}
\lambda=\sum_{j=1}^{n+1}l_je_j=:(l_1,\ldots,l_{n+1})_e=\sum_{i=1}^n\lambda_i\w_i=:(\lambda_1,\ldots,\lambda_n)_\w,
\qquad  \sum_{i=1}^{n+1}l_i=0.\end{gathered}$$

Consider a point $\lambda\in {\mathcal H}$ with coordinates $l_j$ and $\lambda_i$ in the $e$- and $\w$-bases, respectively. Using $\alpha=\mathfrak{C}\w$, i.e. $\w_i=\sum^n_{k=1}(\mathfrak{C}^{-1})_{ik}\alpha_k$, we obtain the relations between $\lambda_i$ and $l_j$: $$\begin{gathered}
l_1=\sum^n_{k=1}\lambda_k\mathfrak{C}^{{-}1}_{k1},\qquad
l_{n{+}1}=-\sum^n_{k=1}\lambda_k\mathfrak{C}^{{-}1}_{kn},
\\
l_j{=}\lambda_{1}(\mathfrak{C}^{{-}1}_{1\,j}{-}\mathfrak{C}^{{-}1}_{1\,j{-}1})
{+}\lambda_{2}(\mathfrak{C}^{{-}1}_{2\,j}{-}\mathfrak{C}^{{-}1}_{2\,j{-}1}){+}\!\cdots\!
{+}\lambda_n(\mathfrak{C}^{{-}1}_{n\,j}{-}\mathfrak{C}^{{-}1}_{n\,j{-}1}),
\quad j=2,\dots,n.
$$ or explicitly, $$\begin{gathered}
\label{l_to_lambda}
\lambda_i=l_i-l_{i+1},\qquad i=1,2,\ldots ,n.\end{gathered}$$ The inverse formulas are much more complicated $$\begin{gathered}
\label{lambda_to_l}
l=A\lambda,\end{gathered}$$ where $l=(l_1,\dots,1_{n+1})$, $\lambda=(\lambda_1,\dots,\lambda_n)$, and $A$ is the $(n{+}1)\times n$ matrix: $$\begin{gathered}
A=\tfrac{1}{n+1}\left(
\begin{smallmatrix}
n&n-1&n-2&\cdots&2&1\\
{-}1&n-1&n-2&\cdots&2&1\\
{-}1&{-}2&n-2&\cdots&2&1\\[-1 ex]
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
{-}1&{-}2&{-}3&\cdots&{-}(n-1)&1\\
{-}1&{-}2&{-}3&\cdots&{-}(n-1)&{-}n
\end{smallmatrix}
\right).\end{gathered}$$

The Weyl group of $A_n$ {#ssec_Weyl_group}
-----------------------

The Weyl group $W(A_n)$ of order $(n+1)!$ acts in ${\mathcal H}$ by permuting coordinates in the $e$-basis, i.e. as the group ${\rm S}_{n+1}$. Indeed, let $r_i$, $1\le i\le n$ be the generating elements of $W(A_n)$, i.e, reflections with respect to the hyperplanes perpendicular to $\alpha_i$ and passing through the origin. Let $x=\sum\limits_{k=1}^{n+1}x_ke_k=(x_1,x_2,\dots,x_{n+1})_e$ and 