the set of vectors $q=\sum_i^n p_{i}\omega_{i}$, $p_{i}\in\mathbb{Z}$. The reflection generator with respect to the hyperplane orthogonal to the simple root $\alpha_{i}$ is $r_{i}$, $(i=1,...,n)$, which operates on an arbitrary vector $\lambda$ as $$r_{i}\lambda=\lambda-\frac{2(\lambda,\alpha_{i})}{(\alpha_{i},\alpha_{i})}\alpha_{i}.\label{eq:Weylreflection}$$ It transforms a fundamental weight vector as $r_{i}\omega_{j}=\omega_{j}-\alpha_{i}\delta_{ij}$. These generators form the Coxeter reflection group (Weyl group) acting on the root system $G$ defined by the presentation $W(G)=\langle r_{1},...,r_{n}|(r_{i}r_{j})^{m_{ij}}=1\rangle$. When they act on one of the simple roots of a simple Lie algebra they generate the root system $W(G)\alpha_{i}$.

The concept of polytopes is important to our discussion. The root polytope is a convex polytope whose vertices are vectors of the root system. The highest weight vector for an irreducible representation of the Lie algebra is defined as the weight vector $\omega_{rep}=(p_{1},...,p_{n})$, where $p_{i}$ are positive integers called Dynkin labels. The highest weight of any irreducible representation decomposes on fundamental weights, and its components are its Dynkin labels $$\omega_{rep}=\sum_{i}p_{i}\omega_{i}.\label{eq:highestweightdecomposition}$$ The highest weight vector has an associated polytope, which is the convex polytope possessing the symmetry of the Coxeter-Weyl group as the orbit of this highest weight vector $W(G)(p_{1},...,p_{n})=(p_{1},...,p_{n})_{g}$. With this notation the root polytope of $W(A_{n})$ is given, in a simplified form, by $(10...01)_{A_{n}}$. For $A_{2}$, for 