  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.
