Groups are abstract mathematical structures that are fundamental to nearly all branches of abstract algebra. Representation theory gives us the tools to study groups using concrete matrices. For this reason representation theory is a powerful tool, but finding representations is not an easy task. In this talk we shall explore some of the fundamentals of representation theory. We will use this foundation to show that the set of degreed polynomials in two variables is an irreducible GL2 module. This result can then be generalized to our main theorem, which constructs a large class of irreducible representations of the general linear group.