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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.

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  • $\begingroup$ I'm not positive, but would this earlier answer by yours truly help you here? Assuming that you want to study the orbit of the weight under the Weyl group rather then the orbit of a vector in $R$. $\endgroup$ – Jyrki Lahtonen Jul 17 '15 at 16:41
  • $\begingroup$ I am not sure I understand your issue with your idea. You are looking for the orbit of the vector, not the span of the orbit. Just because the orbit spans R doesn't mean you have done anything wrong. $\endgroup$ – WSL Jul 17 '15 at 17:12
  • $\begingroup$ And just to make sure. Not all vectors in $R$ are weight vectors (assuming a semisimple Lie group so that it makes sense to talk about weight vectors), but they often are linear combinations of weight vectors. $\endgroup$ – Jyrki Lahtonen Jul 17 '15 at 22:01
  • $\begingroup$ @JyrkiLahtonen thanks for your comment. As far as I understand the Weyl orbit is different from the orbit I'm trying to compute. The orbit I'm looking for are all the weights that are connected to my initial weight trough a group transformation $g$ $\endgroup$ – jak Jul 18 '15 at 6:27
  • $\begingroup$ @WSL My problem is that the roots act like ladder operators on a given weight. That's how one can compute the weight system of a given representation in the first place. In turn this means that if I simply compute all the weights that I get from a given weight by acting on it with the roots I always get the complete weight system of the given representation. This would mean the orbit of a weight is always the complete weight system of the representation our weight lives in. $\endgroup$ – jak Jul 18 '15 at 6:30

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