Polynomial tau-functions of the KP, BKP, and the s-component KP hierarchies

Abstract

We show that any polynomial tau-function of the s-component KP and the BKP hierarchies can be interpreted as a zero mode of an appropriate combinatorial generating function.As an application, we obtain explicit formulas for all polynomial tau-functions of these hierarchies in terms of Schur polynomials and Q-Schur polynomials respectively. We also obtain formulas for polynomial tau-functions of the reductions of the s-component KP hierarchy associated to partitions in s parts.

Linear transformations of vertex operator presentations of Hall-Littlewood polynomials

“…In this note we show that specializations considered in Sections 5.1, 5.2 provide one more description of all polynomial tau-functions of the KP and the BKP hierarchies. This observation will easily follow from the results of [27] with the advantage that the new formulation immediately implies that a number of Schur-like symmetric functions that can be found in the literature provide polynomial tau-functions of the KP hierarchy (when these symmetric functions are expressed as polynomials in power sums). For example, we recover as a particular case our earlier result [47] that multiparameter Schur Q-functions are tau-functions of the BKP hierarchy.…”

“…We review properties of symmetric functions following [33,50]. The setup is similar to [22,27,43,47]. Consider the algebra of formal power series C…”

“…In [24,25,26] it is demonstrated that any polynomial tau-function of the KP-hierarchy is obtained from a Schur polynomial by certain shifts of arguments in the Jacobi-Trudi formula, and that any polynomial tau-function of the BKP or the DKP hierarchy is described by a shift of arguments in the Pfaffian formula for Schur Q-polynomials. Another approach in [27] proves that any polynomial tau-function of the KP, the BKP and the s-component KP hierarchy can be interpreted as a zero-mode of an appropriate combinatorial generating function.…”

“…The initial motivation for this project was the study of tau-functions of the KP and the BKP hierarchy [6,7,8,9,19,49] by generalizing the methods of [47], where the author proved that the multiparameter Schur Q-functions are tau-functions of the BKP hierarchy. We aimed to provide a description of all polynomial tau-functions of the KP and the BKP hierarchies unifying the ideas of [47] and [27]. This is done in Section 6.…”

$$\underline{p}$$-reduced Multicomponent KP Hierarchy and Classical $$\mathcal {W}$$-algebras $$\mathcal {W}(\mathfrak {gl}_N,\underline{p})$$

“…Theorem 2.4. ([KvdL19,Thm.7]). For all choices of the data (2.6), the tau-function τ (t) dened by (2.10)(2.12) is a polynomial solution of the p-KdV bilinear equation (2.2), of charge k as in (2.8).…”