We present a new disaggregated formulation of the Generalized Assignment Problem (GAP), consisting of O(mn2) variables and constraints, where m denotes the number of agents and n the number of jobs. In contrast, the traditional formulation consists of O(mn) variables and constraints. The disaggregated formulation is stronger than the traditional formulation; the linear programming relaxation of the disaggregated formulation provides tighter lower bounds. Furthermore, this new formulation provides additional opportunities for generalizations of the well-known Cover and (1, k)-Configuration inequalities that are not present in the traditional formulation. Under certain restrictive conditions, both inequalities are shown to be facets of the polytope defined by feasible solutions of GAP. We introduce two classes of inequalities involving multiple agents that are specific to this formulation. One class of inequalities is called the Bar-and-Handle (1, ) Inequality, which under certain restrictive condition is a facet of the polytope defined by feasible solutions of GAP. Finally, we introduce another important class of inequality called the 2-Agent Cardinality Matching Inequality involving exactly two agents. Given the un-capacitated version of GAP in which each agent can process all jobs, we first show this inequality to be facets of the polytope defined by the associated bipartite graph. We then show how this inequality can be easily lifted to become a facet of the polytope defined by feasible solutions of GAP. Finally, we show that when m = 2, this inequality, along with trivial facets completely describe the polytope associated with the un-capacitated version of GAP.