# Roots and Weights

I use a Mathematica package to compute roots and weights (and other things) but the package gives me only the expression of the roots in $\omega$-basis (basis of fundamental weights) and in the $\alpha$-basis (basis of the simple roots). And the same for the weights. So my question is :

Let $V$ be a finite dimensional Euclidean vector space and a root system in $V$. As I said suppose I only have the roots expressed in the $\omega$-basis (or $\alpha$-basis) but I haven't the expression of the fundamental weights in an orthonormal basis of $V$ (nor the simple roots). How can I get the roots expressed in $V$ ? I mean, how can I plot this with the previous informations :

• You should have either a Cartan matrix, if the roots come from some Lie algebra construction, or an inner product for which the values of $(\omega_{i}, \omega_{j})$ or $(\alpha_{i}, \alpha_{j})$ are known. That should be enough to plot angles. – Siddharth Venkatesh Apr 19 '14 at 10:28