Polynomial Tau-Functions for the Multicomponent KP Hierarchy

Abstract

In a previous paper we constructed all polynomial tau-functions of the 1-component KP hierarchy, namely, we showed that any such tau-function is obtained from a Schur polynomial $s_{\lambda }(t)$ by certain shifts of arguments. In the present paper we give a simpler proof of this result, using the (1-component) boson–fermion correspondence. Moreover, we show that this approach can be applied to the $s$-component KP hierarchy, using the $s$-component boson–fermion correspondence, finding thereby all its polynomial tau-functions. We also find all polynomial tau-functions for the reduction of the $s$-component KP hierarchy, associated to any partition consisting of $s$ positive parts.