KdV type hierarchies, the string equation and $W_{1+\infty}$ constraints

Abstract

To every partition $n=n_1+n_2+\cdots+n_s$ one can associate a vertex operator realization of the Lie algebras $a_{\infty}$ and $\hat{gl}_n$. Using this construction we make reductions of the $s$--component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. Now assuming that (1) $τ$ is a $τ$--function of the $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy and (2) $τ$ satisfies a `natural' string equation, we prove that $τ$ also satisfies the vacuum constraints of the $W_{1+\infty}$ algebra.