Giambelli type formulae in the BKP hierarchy

TL;DR

This paper explores Giambelli type formulas within the KP and BKP hierarchies, providing an alternative proof approach that extends to the BKP case without relying on Sato's theory.

Contribution

It offers a new proof method for Giambelli formulas in the KP hierarchy and extends this approach to the BKP hierarchy.

Findings

Established equivalence between addition formulae and hierarchy solutions

Provided an alternative proof for Giambelli formulas

Extended proof technique to 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.