While studying representations of Simple Lie Algebras the procedure is quite straight forward.
- Find the Cartan Sub-Algebra
- Label states with the corresponding eigenvalues
- Use the remaining generators to construct SU(2) blocks and use them as raising/lowering operators.
This leds to the notion of roots and in general it's straight forward to find the corresponding root structure or decomposition for the classical groups.
Having all this information (root system, Cartan Matrix, and so on) one can in principle use them to construct new representations of the algebra. My question is the following, given all the information regarding the roots of a classical Lie algebra how can I build the weights of representation with fixed symmetry properties. For example say you are in SP(N) and you know all the root system and Cartan Matrix, how do you build the Weights corresponding to the antisymmetric representation?. Is there a systematic way to do so?.
Any bibliography will be appreciated!