Semi-simple lie groups and their fundamental representations I've got a really basic question on the representation theory of semi-simple Lie groups.  I know that a rank-R semi-simple Lie group possesses R fundamental representations.  But is the relation between semi-simple Lie groups and their fundamental representations injective?  That is, can two distinct semi-simple Lie groups possess the same set of fundamental representations?  
I have always just assumed that the answer was no, but I don't know how to prove it ( in my defense I'm an undergrad physicist, not a mathematician, by trade!)  Any straightening out of this matter would be most appreciated.  
 A: It seems to me that you need to think more about what it is you really want to know.
First, taking your question at face value. It sounds like a version of 20 Questions that starts with "I'm thinking of a semisimple Lie algebra". It is not clear what questions I am allowed to ask. My overall impression is that we could have an involved discussion about the rules and arrive at the point where I have a set of questions you find acceptable and which allow me to determine the Lie algebra. For example, I find it plausible that the rank and the list of dimensions of fundamental representations determine the Lie algebra. I don't feel inclined to attempt a proof. Any such proof would rely heavily on the classification of simple Lie algebras and tricks involving the lists of dimensions of fundamental representations.
I would also question whether this is the right question from the point of view of the physics. First we don't know how many fermions there are. All we can say is that we have done these scattering experiments at these energies and this is the list of particles we have seen. Second if we think we have found all particles then the gauge bosons form the adjoint representation so we know the dimension of the adjoint representation. Is this information you would disclose in the game of 20 Questions? Thirdly I don't know of any reason why the fermions should be a fundamental representation.
As I understand it supersymmetry does impose strong conditions on the representations. My understanding is that there are good physical arguments for restricting the spin to be at most 2 (or maybe less?). I would be interested in seeing these various physical conditions listed and it would then be  challenging problem to classify the solutions. This must be known in the physics community.
