The generalized Giambelli formula and polynomial KP and CKP tau-functions

TL;DR

This paper introduces generalized Giambelli and Jacobi-Trudy formulas to describe all polynomial tau-functions of the KP hierarchy and extends these results to the CKP hierarchy and its reductions, including the Kaup-Kupershmidt hierarchy.

Contribution

It provides new generalized formulas for polynomial tau-functions and applies them to classify solutions of CKP and related hierarchies.

Findings

Derived generalized Giambelli and Jacobi-Trudy formulas for polynomial tau-functions.

Classified all polynomial tau-functions of the CKP hierarchy.

Identified all polynomial tau-functions of the Kaup-Kupershmidt hierarchy for n=3.

Abstract

The first part of the paper is devoted to two descriptions of all polynomial tau-functions of the KP hierarchy: by a generalized Jacobi-Trudy formula, and a generalized Giambelli formula. We use the latter formula in the second part to obtain all polynomial tau-functions of the CKP hierarchy and its n-reductions. In particular, for n=3 we find all polynomial tau-functions of the Kaup-Kupershmidt hierarchy.