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

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. 

Polynomial Tau-Functions of the n-th Sawada–Kotera Hierarchy

“…The CKP hierarchy (KP hierarchy of type C) can be constructed by making use of the KP hierarchy, and assuming the additional constraint L( t, ∂) * = −L( t, ∂) (see, e.g., [5] for details). Its 3-reduction is defined by the constraint that L( t, ∂) = L( t, ∂) 3 is a differential operator, and the corresponding hierarchy is…”

“…For k = 5, we obtain the Kaup-Kupershmidt equation, the simplest non-trivial equation in this hierarchy. All polynomial tau-functions of ( 9) (and all n reductions of the CKP hierarchy) have been constructed in [5].…”