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 $gl_\infty$ and hence a $\tau$ function. We use these fundamental representations to study the Gauss decomposition in the infinite matrix group $Gl_\infty$ and to express the Baker function in terms of $\tau$-functions. The reduction to loop algebras is discussed.