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 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 -component KP hierarchy, using the -component boson–fermion correspondence, finding thereby all its polynomial tau-functions. We also find all polynomial tau-functions for the reduction of the -component KP hierarchy, associated to any partition consisting of positive parts.
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Victor G. Kac, Johan W. van de Leur, Polynomial Tau-Functions for the Multicomponent KP Hierarchy. Publ. Res. Inst. Math. Sci. 58 (2022), no. 1, pp. 1–19DOI 10.4171/PRIMS/58-1-1