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

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 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 $τ$--function are equivalent (1) $τ$ is a $τ$--function of the $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy which satisfies string equation, $L_{-1}τ=0$, (2) $τ$ satisfies the vacuum constraints of the $W_{1+\infty}$ algebra. Talk given at the V International Conference on Mathematical Physics, String Theory and Quantum Gravity at Alushta, June 10-20 1994

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