Giambelli type formulae in the BKP hierarchy

Abstract

In this paper, we study Giambelli type formula in the KP and the BKP hierarchies. Any formal power series $\tau(x)$ can be expanded by the Schur functions. It is known that $\tau(x)$ with $\tau(0)=1$ is a solution of the KP hierarchy if and only if the coefficients of this expansion satisfy Giambelli type formula. It is proved by using Sato's theory of the KP hierarchy. Here we give an alternative proof based on the previously established results on the equivalence of the addition formulae and the KP hierarchy without using Sato's theory. This method of the proof can also be applied to the case of the BKP hierarchy.