Partitions, vertex operator constructions and multi-component KP equations

For every partition of a positive integer n in k parts and every point of an infinite Grassmannian we obtain a solution of the k component differential-difference KP hierarchy and a corresponding Baker function. A partition of n also determines a vertex operator construction of the fundamental representations of the infinite matrix algebra gl_∞ and hence a t function. We use these fundamental representations to study the Gauss decomposition in the infinite matrix group Gl_∞ and to express the Baker function in terms of t-functions. The reduction to loop algebras is discussed.