KdV type hierarchies, The string equation and W1+infinity constraints

To every partition n = n1 + n2 + + ns one can associate a vertex operator realization of the Lie algebras α∞ and . 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 [n1, n2, …, ns]th reduced KP hierarchy and (2) τ satisfies a ‘natural’ string equation, we prove that τ also satisfies the vacuum constraints of the W1+∞ algebra.