I am new to this field and learning this subject on my own so apologies if I interpret anything incorrectly. My question is while working out representation theory of semi-simple Lie groups through roots-weights classification we obtain fundamental representations and express all other representations as tensor product presentations so for example: $SU(3)$ have $(1, 0)$ and $(0, 1)$ as fundamental reps and then we take tensor product of these reps. Now for any arbitrary tensor product representation of these fundamental reps can be reducible so then we tend to employ Young's Tableaux to get direct sum of irreducible reps (which mainly involves all possible cross indices symmetrisation-antisymmetrization schemes for each irr-rep). My first question is:
Can we apply similar analysis of Young's Tableaux for non-compact groups like SO(1, 3), particularity for spinor representations of this group. Can we obtain irr reps of tensor product of fundamental reps using Young's Tableaux.
If yes then what are cartan generators and roots in this case.
If yes then what I understood by analogy is that fundamental reps in this case will be $(1/2, 0)$ and $(0, 1/2)$ which we call as left-hand spinors and right-hand spinors (these terms comes from physics literature and I don't know is it used in mathematics literature too). Now In this case how Young Tableaux will work, for example how can I obtain 4-vector rep as $(1/2, 1/2)$ tensor product rep through Young's Tableaux.
Is there some source where I can read up on it a bit more.
Thank you.