Representation theory can be simply defined as the study of the ways a group acts on a vector space. A large piece of this process involves understanding weight spaces, particularly irreducible highest weight spaces. We will explore weight spaces of irreducible representations by looking at sl(n, C), U_{q}(sl(n)) and U_{q}(ŝl(n)). As the structure of the space becomes more complicated we utilize different combinatorics to understand these irreducible, highest weight spaces.