Geodesic
The $[n_1,n_2,\dots,n_s]$-th reduced KP hierarchy and $W_{1+\infty}$ constraints
Teoretičeskaâ i matematičeskaâ fizika, Tome 104 (1995) no. 1, pp. 32-42

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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. 
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     author = {J. van de Leur},
     title = {The $[n_1,n_2,\dots,n_s]$-th reduced {KP} hierarchy and $W_{1+\infty}$ constraints},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {32--42},
     publisher = {mathdoc},
     volume = {104},
     number = {1},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TMF_1995_104_1_a3/}
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J. van de Leur. The $[n_1,n_2,\dots,n_s]$-th reduced KP hierarchy and $W_{1+\infty}$ constraints. Teoretičeskaâ i matematičeskaâ fizika, Tome 104 (1995) no. 1, pp. 32-42. http://geodesic.mathdoc.fr/item/TMF_1995_104_1_a3/