Topics in Representation Theory: SU(2) Representations and Their Applications

We've so far been studying a specific representation of an arbitrary compact Lie group, the adjoint representation. The roots are the weights of this representation. We would now like to begin the study of arbitary representations and their weights. An arbitrary finite dimensionsional represesentation will have a direct sum decomposition V = α V α where the α are the weights of the representation labelled by elements of t * , and V α is the α-weight space, i.e. the vectors v in V satisfying Hv = α(H)v for H ∈ t. The dimension of V α is called the multiplicity of α. The problem we want to solve for each compact Lie group G is to identify the irreducible representations, computing their weights and multiplicities. An important relation between roots and weights is the following: so the roots act on the set of weights by translation. We will begin with the simplest case, that of G = SU (2). This case is of great importance both as an example of all the phenomena we want to study for higher rank cases, as well as playing a fundamental part itself in the analysis of the general case.