I am particularly interested for $SO(3)$.
Let us say that I start with the natural/defining $3$-real-dimensional vectorial representation of $SO(3)$ and I choose the generator of rotation in the $1$-$2$ plane as my Cartan subalgebra. So I have $3$ weight vectors with weights $1$, $0$, $-1$ w.r.t. this generator.
Then is there a way to see what will be the weights under this chosen Cartan of $SO(3)$ of the weight vectors in the $m$-fold antisymmetrization of this $3$-dimensional vectorial representation?
- It would be great if possible someone can explain the corresponding general result for $SO(n)$, $U(n)$ and $SU(n)$ where too I think the same question can be asked.