The $n$-component KP hierarchy and representation theory

Abstract

Starting from free charged fermions we give equivalent definitions of the $n\/$-component KP hierarchy, in terms of $\tau\/$-functions $\tau_\alpha\/$ (where $\alpha \in M =\/$ root lattice of $sl_n\/$), in terms of $n \times n\/$ matrix valued wave functions $W_\alpha(\alpha\in M)\/$, and in terms of pseudodifferential wave operators $P_\alpha(\alpha\in M)\/$. These imply the deformation and the zero curvature equations. We show that the 2-component KP hierarchy contains the Davey-Stewartson system and the $n\geq3\/$ component KP hierarchy continues the $n\/$-wave interaction equations. This allows us to construct theis solutions.