Let $\Phi$ be a classical root system and let $\omega_i $ be the i-th fundamental weight.
I need to find the orbit of $\omega_i$ by the action of the Weyl Group $W$.
How to compute it? I'm looking in particular at the "simpler" case of the root system $A_n$.
In general a way is find a set of minimal coset representatives of $W / Stab(\omega_i)$ and looking at the images of $\omega_i$ by this elements.
But, again, usually it is not simple find the minimal cosets representative of such a quotient.
Any help or reference is well accepted