The $[n_1,n_2,\dots,n_s]$-th reduced KP hierarchy and $W_{1+\infty}$ constraints

To every partition $n=n_1+n_2+\dots+n_s$ one can associate a vertex operator realization of the Lie algebras $a_{\infty}$ and $\hat{gl}_n$. Using this construction we obtain reductions of the $s$-component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. We show that the following two constraints on a KP $\tau$–function are equivalent (1) $\tau$ is a $\tau$–function of the $[n_1,n_2,\dots ,n_s]$–th reduced KP hierarchy which satisfies string equation, $L_{-1}\tau =0$, (2) $\tau$ satisfies the vacuum constraints of the $W_{1+\infty}$ algebra.