Given some element $\phi$ of a representation $R$ of a group $G$, the orbit $G(\phi)$ of $\phi$ is defined as the set $g \phi \ \forall \ g \in G$.
We can write every element of a given representation as a weight vector $w$. How can I compute the orbit of $w$, i.e. the set weights in this orbit? Specifically my problem is computing $ g w$. How does the group act on the weight vector?
My idea was to act with the roots (that correspond to the generators of the group) on the weights. Unfortunately this way I can get every weight that corresponds to $R$, i.e. the orbit would always be the complete representation $R$. Therefore this must be wrong.